Ezreal is a caster carry. He doesn't win fights very often in straight-up DPS, but does a very good job of poking people down from long range, and finishing them off with caster burst and mobility. His laning phase is relatively weak, but his mid-game spike is stronger than most other AD carries, and his mobility allows him to control fights incredibly well into the late game.

Items:
Early game:
Doran's Blade is the ideal opening on Ezreal, as you usually want to play fairly safely until you can buy a few core items, and farm it out from long range. Ezreal also has incredibly low base HP, so opening Long Sword and 3x Health Potions for the extra sustain is pretty dangerous.
Tear of the Goddess is your highest priority on Ezreal. Try to find an opportunity to buy it sooner rather than later, so you can begin stacking it, and start trading in lane with Mystic Shot.
Cull is a decent buy on Ezreal early, if you base with an odd amount of gold. It'll help hold you over and keep you healthy until you can buy bigger items.
Manamune is your priority item. It'll provide you with a spike in damage, that'll bring you back into competition in lane, and provides incredible damage late game.
Sheen is your second priority, as it allows you to deal some serious damage with Mystic Shot in lane.
Phage is your third priority, as it lets you control fights with additional mobility, and gives you some health and AD for trading.
Ionian Boots of Lucidity are the ideal boots on Ezreal, as they provide him added mobility through Arcane Shift cooldown, and added threat, as abilities are a large part of Ezreal's damage.
Core:
Muramana is your mid-game damage spike on Ezreal. It allows Mystic Shot to deal substantial poke, and your auto attacks are put back on par, if not slightly above those of standard auto attack carries, for a short time.
Trinity Force is Ezreal's bread and butter. It provides even more Mystic Shot damage, added health, AD, and attack speed for trading, mobility, and CDR (for more damage and mobility).
Blade of the Ruined King is yet another DPS and Mystic Shot poke spike on Ezreal. It also provides sustain, and catching power through the active.
Lord Dominik's Regards allows Ezreal to DPS down front liners into late game fights.
Other Items:
Maw of Malmortius is a great final item, as it caps CDR, and allows Ezreal to survive magic damage burst. The Lifegrip passive will also allow him to heal back after taking burst, mid-fight.
Mercurial Scimitar is your 'oh-sh*t' button. Build one against teams with deadly lock-downs.
Mortal Reminder is your substitute for Lord Dominik's Regards, against heavy healing.
Sterak's Gage is your Maw of Malmortius alternative for AD assassins.
Bloodthirster is another option if you're scared of getting killed by burst. Build it instead of Blade of the Ruined King for lower DPS, but higher sustain, and a shield.
Iceborn Gauntlet is a low damage kiting alternative to Trinity Force. Only play this with a kite comp, or against a full AD team, if your team has additional DPS champions.
Guardian Angel If you're snowballing and want to end the game with an aggressive 6 item push, consider this.