Jax is a AP/AD Hybrid champion who works well when rolled as an assassin/dps/carry, though the team shouldn't rely souly on him to fulfill all these rolls at the same time. As the term "hybrid" implies, he is GOOD at all of these jobs, but clearly not the best. His ability to adapt to multiple scenarios during games makes him a powerful member of any 5v5 team, as well as a dominating force during the 3v3.

I consider this guide to be an update of earlier guides for Jax. Although this build tends to be slightly different in some regards, much of his play style has remained the same after the more recent patches, so we may notice some major similarities in this guide as well. I've played Jax for quite a while now and have found that this has been by far my most effective build for the way that I play him, so feel free to leave some constructive feedback in the comments section.

Marks: There are a few ways you could run with this. Personally, I have found that the higher magic penetration comes in handy, seeing as your spike damage is going to be coming from using your abilities, all of which deal magic damage. However, seeing as Jax is also an auto-attacking fiend, it certainly would not be a bad idea to invest in critical chance (Mark of Malice). Crit chance will also work well with your Quints as well since they are Crit Damage. Another choice I'd consider would be Strength, seeing as this will enable you to harass and start racking up kills early early game, as opposed to building slowly into freight train of death.

Seals: This is a no-brainer: Jax needs to dodge in order to get his stun to proc. Naturally we will need to increase his chances of dodging. In addition to allowing him to use Counter-strike, Seals of Evasion will also increase his survivability against auto-attackers and people who don't know how to there skill shots and/or AOE. Many a gank has been thwarted because no-one could actually get more than 2 or 3 shots on me as I run away, turn and /dance in front of my tower.

Glyphs: I'll stand by AP late-game, as this will give you enough momentum to keep the enemy on edge throughout the entire game. Some people have told me I am an idiot for thinking that this is actually helpful, seeing as Jax gets such a healthy dose from his items anyway by endgame, a comment to which the proper rebuttal is a well aimed Empowered Leap Strike to noggin with all 27.63 extra AP from your Glyphs of Force. (I applied this in a literal sense once, albeit minus the Empower and runes: needless to say that someone has not commented on how poorly I choose my runes since)

Quintessence: As mentioned in the Marks section, I like to use my Quintessence of Furor. A valuable part of Jax's attributes will be his ability to inflict high amounts of spike damage, and I see no reason why this should not also apply to his auto-attack, seeing as his attack speed increases the longer he stays in a brawl (more hits = more chances for crit). A big crit can often turn the tide in a brawl, and it's always nice to be able to occasionally one shot enemy minions in the early game.

I won't go into to much detail with spells, seeing as everyone knows what they all do and can kinda guess which ones might be good. Instead, I'll explain why I continue to choose mine and list a few that might be useful to others.

Heal: Heal rocks. Especially when going 1v1 in the brawl, your heal will trick opponents into thinking that you have no chance of winning the brawl that you initiated. In the early game, try to hold off on using this until you are getting your kill.
Ignite: You can't always chase a person down, so drop the ignite on them to burn them down the rest of the way. 60% of the time, it works. Every time.

I don't use them, but I see why some people might:
Exhaust
Ghost (I'd recommend over flash)
Flash
Cleanse

For my masteries, I think it would be easier for me to give you a "Why not" list than reading the excruciatingly detailed explanations of why each point was placed where I put it. Sooo...

Why not:
Sorcery? While shorter cooldowns are helpful, Jax has some fairly short cooldowns to start off. In most cases, if you feel like the cooldown is too long, you are more than likely casting it too early, e.g., you were unable to escape a mass of enemy champions because you wanted to try and kill someone with your Leap Strike.
Resistance? Because you walk all over casters already. Your "magic resistance" lies in the fact that they can't get any spells off while they are getting clobbered and stunned.
Anything in the Utility tree?- As hard as it may be to admit, Jax is not invincible. As a matter of fact, he is actually quite squishy and needs the extra survivability. Also, seeing as Nimbleness is also activated when you dodge (noticing a pattern here?), it is better to sacrifice this whole tree just to get that extra bit of speed to cut in or chase down for a kill.

Boots of Speed: being able to move faster than a caster can account for is an important part of Jax's survivability/killability in the early game. With just about every single ranged AOE attack that the enemy has, you will have about 1/2 a second to quickly change directions and avoid taking the damage- this speed boost will be what saves you 90% of the time. From there, you simply turn about and harass them, seeing as you can easily catch them with your higher movement speed and death-from-above Leap Strike.

Health Potions: Because despite your best efforts, you can't avoid every bit of damage. Also, it's one of the only useful things to buy after you buy your boots. This will help you stay in lane longer than your opponent, meaning more money and faster leveling.

Vampiric Scepter: I am a strong advocate of this item for any DPS type character without built in lifesteal (I don't really think that lifesteal is something anyone would say isn't important), and it remains true with Jax. Stay in lane longer, or back off and heal up on some jungle neutrals so that you can get back in lane/team fight quickly.

Guinsoo's Rageblade: For obvious reasons, this item syncs well with Jax. Not only will you have the attack speed boost from your ult passive, you get the same effect from this, plus extra ability power (which, might I add, also gives Jax health via his passive). I quite honestly believe that this ability was made specifically for Jax.

Bilgewater Cutlass: Extra lifesteal and AD(which also gives you health), plus a handy, short-range damaging slow with a 60 sec cooldown for following through on a carry or frustrating the enemy carry.

Hextech Gunblade: Increased everything from Bilgewater Cutlass, plus lots of spellvamp and AP. This is the point in a game when people begin to despise yet fear you.

Phage: The sole reason for this is to pick up that on-hit slow. CAUTION: you may notice a change in the enemy mindset-- when they know they can't run from you once you've caught up, they'll run from you when you show up.

Atma's Impaler: The passive on this item increases your AD according to your health (of which you have a big basketful at this point). And of course that extra AD will further increase your health (a shame that it won't go on into an infinite loop, right?).

Lichbane: Every time you cast a spell, your next auto attack will deal 100% of your AP as damage. When set off by Empower, you come close to taking out half the health of most brawlers.

Phantom Dancer: Feeling froggy? Drop your boots and pick up the Phantom Dancer. You keep your movement speed high while trading off that extra dodge rating for some extra attack speed and Crit damage.

No matter what anyone else says, start off with your Leap Strike. If anyone tells you differently, they are either screwing with you or are too stubborn to admit they're wrong.

With that out of the way, we move on to lvl 2 and 3. Pick either Empower or Counter-Strike based on the person or people you are laning against: if they are pushing, get your Counter-Strike first so that you have something to stop the momentum with while giving yourself a chance to not take tons of damage with. Alternatively, if you are pressing hard, grab your Empower so that you can do a bit of extra damage as you harass them into the danger zone and away from that precious Exp.

From here on out, its cut-and-dry: fill Empower, then Leap Strike, and last Counter-Strike, putting points in Relentless Assault whenever possible.

Why Empower and not Leap Strike? While you will be able to do more damage with every LS/E combo when leveling LS first, remember that Empower is not just for jumping at people. Being able to brawl when needed is very important, and if your lifesteal is low, you're going to need every point of damage you can get as soon as you can get it.

A brief note on why I consider it silly to level Counter Strike first: this ability only increases in dodge chance and magic damage, and not by much. If one day Riot decides that this should also increase the duration (I find the very thought titillating), expect an update within minutes.

If I had to generalize Jax's strategy, I'd say it is keeping up a strong bluff. In most cases, you will be using deceitful tactics that make you look like a bad player to get your early kills, or you will be popping off huge spike damage to trick people into backing off when you don't have nearly enough oomf to follow through if they were to make a stand.

Jax has 2 ways of accomplishing this goal of maintaining an effective poker face. The first is with his Empower/Leap Strike combo. When you are being pressed by an opponent, you will find an effective stopping mechanism is your lamp firmly applied to their forehead. Generally, a player will see that they have gone from full to 66% in .01 seconds and will seriously reconsider whether or not they should take you on. This is your bluff- in many cases early on, you simply don't have the AD yet to stay and fight them. If they turn to run, do your best to take some minion fire to dodge so you can stun your fleeing adversary, further demoralizing them.
The second way in which you will be able to trick your opponent into his demise is when you DO have enough AD to woop em. This is where your extra crit damage and increasing attack speed will come into play: they may be ahead of you at first, but they will quickly see that, as your attack speed grows, you start gaining back more and more health from lifesteal. Combined with those random spiking crits, expect many curses to fly your way after you mop the floor with their face.

Harassing (you should be doing this a lot, so make real friendly with your W-Q):
Empower(W)-Leap Strike(Q)-Counter Strike(E)-back off

Killing/Carrying:
Empower(W)-Leap Strike(Q)-Counter Strike(E)-Auto attack-Cutlass/Gunblade-Empower-Counter Strike-Auto-Leap Strike... If it takes much longer, you probably should not have attacked this champion

Soloing/1v1:
This is where Jax was meant to be. If you are playing a 3v3, be very insistent on getting the top, as Jax does not play nearly as easily on the bottom. In 5v5, this may seem an unusual choice, but Jax is generally able to frustrate the majority of would-be mid-laners. He is able to solo well against many ranged champs since he is able to harass them early on, in many cases dominating the lane and pushing them away from all the EXP entirely after he hits level 3.

It may be wise to start off a bit defensively until the opportunity for an aggressive push opens up. You may be forced to sit back and just watch your minions do all the work if the enemy wants to block you out. His time will come...
When the enemy over-extends, Leap Strike and press him back. If you have been able to invest in Counter Strike at this point, make sure there are SOME(not a horde) of minions around the enemy so that you can soak up some damage from them until you can pop off your stun on the enemy. This will make him seriously wish he hadn't been stuck laning with you. Back off as needed, but make sure that you are pushing further into your minions every time.
Eventually, you will have pushed the enemy out of range of EXP. This will frustrate him, possibly into making a silly mistake. Silly mistakes = kills for Jax or them being forced to port to base, so be ready to take advantage of the opponents leave of senses.
Be sure to be making use of your Health Potions: I generally use them any time my health gets to be around 50% in the early game. This makes you look buff and intimidating. Not to mention how cool you look with that green pixie dust flying off you.

A quick list of Champs Jax struggles with on the solo lane:

Fiddlesticks
Heimerdinger
Nasus
Nocturne
Mordekaiser
Sivir
Kennen

With a partner/2v2:
If you are on a 3v3, get your hands out of that cheeto bag because Jax just got 4 times more labor intensive. On a 5v5, the early game will be tough, but not as difficult since you won't be constantly harassed by the Solo.

Focus on getting your level up and making last hits on minions- playing defense with Jax may be frustrating, but you may find yourself using your Leap Strike on minions more often than enemy champs, at least for a bit.
When you are less tanky than your partner, remember that you need to give them support and open up opportunities for them. This doesn't mean charge in, soak up all the damage, and expect your compadre to finish off all the baddies with you- snipe over-extenders and do your best to guard your buddy from getting buggered up.
Preferably, buddy up with someone who is heavier than you so that you can assume your rightful position- the crash of thunder after the lighting breaks every one apart. DO NOT INITIATE!!! The only time you initiate is when you KNOW you can get out without taking much damage: this is not very common when the enemies are drooling on their keyboards over your slender figure.
Be looking out for ganks: you are a fairly light champ and will generally not make it out if you get bogged down by slows.

The above is mostly useful up to levels 6 or 9. See the bottom of this chapter for some tips on what to do next, as you will be moving around a decent amount after this.

Mid-game:
You may find yourself lane-swapping a decent amount. Depending on how well you did early game, consider whether or not you should move with one or both of your comrades. At this point, you are looking for kills on the lone-wolves of the other teams. Whenever possible, stick with a lane that has no teammates in it to farm some minions and level up.
Pop out of lane to grab dragon as early as level 11, depending on how much lifesteal you have. I say 11 because this is when you can solo it without breaking too much of a sweat.

Late-game:
If your team is on defense, stick with your team (obviously). Otherwise, be moving from lane to lane, keeping constant pressure on the other team. Regroup with your team to take down tanks, but otherwise you will normally be able to solo anyone else (this rule of thumb does not apply to Master Yi, Olaf, or Tryndamere: exercise caution when considering whether or not to solo them. Also, look out for DPS enemies with Sword of the Divine, as this will negate your dodge)

End-game:
Walk forward.

Pros:

High spike damage
The enemy team fears you when you are fed
High lifesteal
Extremely versatile
Purple is the color of royalty
Laugh heartily while the other team says that Jax is easy to play and OP

Cons:

Squishy
Can be bogged down by slows and ganged up on (like most champs...)
^^Throat may become sore from laughing (people talk a lot of trash about my man Jax)

You have chosen to embark on a rewarding endeavor. Having played Jax for a while now, I have found that he is easy to learn, difficult to master. While this build is not perfect and will not make you unkillable, it will bring you a long way towards a high Kill:Death ratio and many victories.

Bare with me on formatting errors as this is my first guide, but I respectfully ask that if you down-vote, leave a comment so I know the areas that this guide might be improved in.