In this guide, I will cover my playstyle of Jax, which is a Tanky AD style. I will cover:

Item Build
Masteries
Runes
Summoner Spells
Skills
Team Work
Summary

Okay, first and foremost, the way I play Jax is of course the way you should play ANY tanky AD character, Atmogs - Atma's Impaler and a Warmog's Armor. I also add a Frozen Mallet in to the core build because it helps with both tankiness, and adds a slow to your basic attacks, so Leap Stirke -> Empower -> Hit will do around 200-300 damage depending on their armor, and then add a slow so you can stay caught up to them until your CD's are up.

Allow me to explain some of the situational items to build with Jax:

One of my favorites to have is Rylai's Crystal Scepter, why? YOU GET A SLOW ON YOUR SPELLS. That means that Leap Strike will slow an an enemy, and then a hit with Frozen Mallet will slow them EVEN MORE, making for a very easy to chase champion, regardless of their buffs, i'm talking to you, Teemo.

Another great item to have is a Force of Nature, why? 76 MR and health regen on top of Warmog's? Yes please. That simple

Always remember, if you are facing an AP heavy team, always grab a Banshees Veil fairly early on to keep that tibbers stun from killing you in a team fight.

More viable situationals:

Frozen Heart - Tankiness + Mana + CDR + EVEN MORE SLOW = Epic
Thornmail - Show that Tryndamere who's boss without even lifting a finger
Sunfire Cape - Jax needs AOE, now he can have some
Spirit Visage - Only if you feel that you need the regen
Black Cleaver - ARP + Attack Speed. 'Nuff said.
Bloodthirster - If you really need lifesteal, I usually lean away from lifesteal
Bloodrazor - Usually not a great item for Jax, but trust me, Mordekaiser and Jarvan will feel it
Phantom Dancer - I don't usually take it, but it would be nice for the attack speed alone
Infinity Edge - More or less a great AD strict item, not as great for Tanky AD unless really late in the game
Trinity Force - I cannot stress how great this item is, you get the Sheen proc x 1.5 + Health and to top it all of, phage proc. I'll sometimes take this over Frozen Mallet if my early game is going particularly well.

Boots - I usually take boots either between Regrowth pendant and Giant's Belt, or I buy level 1 boots first and build into level 2 after Warmog's. So, what kind of boots are good for Jax?
Ninja Tabi - Gives you the armor and Dodge Chance, Oh-So amazing
Merc Treds - Good if you are facing a CC heavy team, like Galio, Amumu or Nasus, but not particularly amazing otherwise
Boots of Swiftness/Boots of Mobility - Good for running between fights, but not particularly useful inside of a fight

Marks - I take Greater Mark of Desolation for the ARP against tanks, but some other good choices are:

Greater Mark of Alacrity - Kind of useful if you don't build other attack speed items
Greater Mark of Malice - Makes for good crits early game before Atma's, but become less useful later game
Greater Mark of Fortitude/Resilience - Probably the most viable picks over Desolation, but i'd still prefer the ARP over health and armor in the early game

Seals - I take Greater Seal of Resilience because having a little extra armor early game can really be a game-changer. Other viable choices are:

Greater Seal of Furor - Adds a little damage to those early crits, makes for good last-hits on minions and champions
Greater Seal of Evasion - Works will with Counter-Strike at early levels
Greater Seal of Fortitude - Most viable other pick, but I think the armor works out better

Glyphs - I take Greater Glyph of Alacrity just because the AS helps alot early game, and with your Ulti, other great and just as viable picks are:

Greater Glyph of Resilience/Fortitude - Armor/Health, same reasons as above
Greater Glyph of Warding - Gives you an edge against early game AP attackers
Greater Glyph of Shielding - Same as above, except for late game.

Quintessences - I take Greater Quint of Evasion for the dodge chance synergy of Counter Strike other viable picks are:

Greater Quint of Resilience/Fortitude - Same reasons as always
Greater Quint of Shielding/Warding - Same as Glyphs
Greater Quint of Desolation - Great for Anti-tank late game *Most Viable*
Greater Quint of Malice - Great for early game last-hitting and lucky kills

I will work more on this guide when I get the chance.

Fairly straightforward, I like 21-9-0 because it allows for alot of extra damage and a bit of tankiness added on. If you prefer ignite over exhaust, take a point out of sunder, or anything else really, and add it to burning embers, use the point fro exhaust in archmage's savvy.

Flash - Basically a get out of jail free card, it can help you escape from almost any situation, almost always take flash over everything, unless you are playing Kassadin.

Viable second picks:

Exhaust - Great for ganks and disabling certain champions for a few seconds in a team fight, pairs well with flash for getaways
Ignite - Good to get those early kills, loses much of its effect late game
Teleport - An okay pick to keep experience going if you die/bluepill, good for jumping around lanes later game, but definitely not my first pick, save it for Singed
Cleanse - Basically another get out of jail free card to break away from that dreaded Sion stun or Taric stun, but otherwise useless late game
Ghost - If you really need to be somewhere or get away from something, ghost is really my only other pick besides exhaust

DO NOT TAKE:

Heal/Clarity - Okay early game but useless in late game
Clarivoyance - Save it for the supports
Smite - For junglers only
Revive - Hopefully you don't die enough to need to revive

I absolutely love the synergy of Jax's abilities.
Leap Strike -> Empower -> Counter Strike (if up) -> Smack them about while stunned -> Repeat
That right there is Jax's bread and butter, so many kills on squishy little champions only take a few seconds and a fraction of your mana, so easy.

Who to lane with:
Rammus - Powerball stun makes for a great initiation, most of my games that I performed well in were while I was laning with my best buddy viceT and his Rammus skills
Sion - Ranged Stun
Taric - Ranged Stun
Veigar - Stun
Annie - Stun
See a pattern, anyone who has a stun and knows how to use it are perfect for laning with Jax

Who not to lane with:
Valdimir - If he pools away, you are stuck and you can't leap strike to him.
Katarina - Bounces around too much with shunpo, plus, no stuns
Cho'Gath - Yes he has a stun, but if he gets fed, you will get absolutely 0 kills in lane, you need to be fed for this build to work to 100%
Ezreal - No stuns, long range = No leap strike to get away.

As a champion, Jax is one of the best, and when paired with an amazing champion with a good stun (RAMMUS) he is even better. He has the potential to do so much unseen damage, but no one ever builds him Atmog's because of his passive, why build health when you could build straight AD? Atmog's will give you the health from warmogs, and the snowball effect if you farm, then Atmas gives you AD for your health, which is the opposite of his passive, so whatever you build, Health = More Damage from Atmas, More damage = more health from passive, and vice-versa, these I think work so well in synergy and are the perfect mix for any good Jax.