Jax can be built in many ways but I feel that the build above is solid no matter what the situation. While Jax is amazing 1v1 he can die to fast to magic bursts so some magic resistance is needed for teamfights since his ult active won't do enough by itself.

Last Updated 11/28/2011

Relentless Assault: Allows you to get hp from both AD and AP allowing Jax to be build versatile and makes him tanky.

Leap Strike: Leap at target ally or enemy to deal dmg benefits from both AD and AP

Empower: Makes your next auto attack do additional dmg benefits from both AD and AP

Counter Strike: Passively increases you dodge chance and when you do dodge you can use an AOE stun that will be available for 7 seconds. Benefits from AP.

Relentless Assault: Lets jax build up attack speed up to a max of 10 stacks with auto attacks along with additional dmg every third attack at rank 3 it goes to a max of 140% attack speed which is just ridiculous especially with guinsoo's rageblade and other attack speed items. Benefits from AP and attack speed to build the passive faster.

Keep in mind that all of the skills do magical dmg which makes Jax very hard to counter damage wise and he can see what your building and customize his build. As you can see that Leap Strike and Empower will do massive dmg no matter what you build AD, AP, Hybrid it will do a lot of dmg.

It more depends on your play style for me its
Relentless Assault-Empower-Leap Strike-Counter Strike.
With this sequence you get more straight up dmg from empower and since empower costs little mana you can last hit easily with it. The first rank of Counter Strike is significant but the rest really isn't too much to bother with until the last 4 lvls. I'm positive most would take leap strike over empower mainly because the utility and the ability to harass more frequently but keep in mind your burst will do more if your empower is higher.

Reds: Attack dmg per lvl is good and will increase your health and dmg to help your leap-empower combo.
Yellow: Attack speed helps you build up your ult stacks faster and do more sustained dmg.
Blue: Ability powere per lvl is good and will increase your health and dmg to help your leap-empower combo.
Quints: Same as Red

You can go flat for a better early game but I like to be at maximum efficiency late game where jax will thrive.

28/2/0 Now that the masteries are reworked I feel that this provides enough to make Jax a pretty damn good dmg dealer as well as add some hp to him from his passive.

Surge: Really good spell that adds ability power and attack speed that works great with jax especially in combination with his ult.

Ignite: Good for securing kills or countering strong healing champs like Fiddlesticks or Dr.Mundo

Flash: This spell is too important and great for escaping or getting in a leap strike to finish off a kill.

Exhaust: To stay on your target or use to slow enemies that are chasing you or in general make you unbeatable 1v1.

Ghost: Great escaping and chasing

Cleanse: Help with cc that is used against you recommended against cc heavy teams.

Teleport: Nice to have and good for map mobility along with teleporting to wards that you or your support has placed to join important fights.

I'll just give a generalization of what the items can do for jax because his build can vary depending on the situation.

Doran's Blade : Basically 130 hp because of passive so makes you tanky early game with minor lifesteal and AD to add to your burst.

Doran's Ring : Same hp increase AP instead for a slightly bigger burst along with mana regen that lets you spam your abilities more often but no lifesteal for additional sustain.

The three core items in this jax build:

Ninja Tabi : his nice pair of boots that give him armor and dodge.

Hextech Gunblade Gives AP,AD,lifesteal,spellvamp, and a ranged slow mini-nuke active that's great for bursting and chasing.

Guinsoo's Rageblade Gives AP,AD, and a unique passive that lets you build your attack speed even faster along with AP.

The items that should be built after the core:

Trinity Force : This is an all around great item for jax built from sheen to add to burst dmg along with a lot of stats that just make jax a monster but expensive.

Atma's Impaler : Gives armor along with a boost to AD based on jax's max hp which in return gives him even more hp :)

Force of Nature : Gives needed magic resist that Jax will need to survive in a fight along with good hp regen based on jax's max hp also just like Atmas.

Banshee's Veil : Same as Force of Nature except less magic resistance and hp regen but a nice spell shield with base health and mana.

The Bloodthirster : If you feel the need to do some sustained dmg and with this you can simply walk out of fight when weak auto attack 2 minions and walk back in at full hp and the AD given will give you up to 300 hp! Usually get this when a Veigar is on the other team to avoid getting some AP because then his burst would hurt you more.

Rylai's Crystal Scepter : Usually you would get this if you aren't going for trinity but its got a nice slow along with AP and health giving you a total of 660 hp.

Void Staff : If you are going AP and see enemies stacking magic resistance that is not the tank in order to get past 40% of their defenses.

If you feel that you are completely dominating you can pick up both Sword of the Occult and Mejai's Soulstealer for ridiculous snowballing increasing you dmg and health immensely.

In general your job in laning is to get as much farm as possible and keep the enemy from getting farm. By team fights you should focus the hard carries ASAP after the fight has initiated to stop them from doing dmg.

9/29/2011: Updated the format to look a lot better.
11/28/2011: Updated a lot to deal with the big changes to masteries.

I hope you liked this guide and use at least some advice from it to play jax well but it is mainly to understand that jax can build to suit the situation in plenty of ways.