highest damage ranged burst
Annie

good melee burst
Kassadin
Poppy
Chogath
Akali

What kind of role are you looking to perform? There are lots of champions to choose from in terms of burst, but most champions have specialized roles, which makes it hard to define a "best" burster.

Kassadin, Akali, and Annie are good gankers
Annie is an extremely effective ganker and has low cooldowns. Her ability to kill opposing champions is completely reliant on your ability to combo your spells and use stun procs to your advantage.
Akali relies on jungling to be a good ganker, if that's what floats your boat, if you're a lower level, it's probably not a good idea to jungle Akali.
Kassadin can close gaps quickly between him and his target, and has a strong escape, making him a reliable ganker
Amumu Rammus and Malphite have reliable mid game burst combos and ganking ability but should fill their role as tanks late game.

Nidalee, Pantheon, Twitch, and Evelynn have strong "combo" and ganking abilities.

However, Nidalee has a high skill ceiling and learning curve, people will hate you for playing Evelynn. Pantheon has poor late game scaling, and Twitch is expensive, but you should keep him in mind, especially if that's not a problem for you.

I'd say twitch or annie. I think twitch is better if you play smart.

poppy has some insane burst if you have trinity force or sheen

Poppy, Kassadin and Akali would be my picks. They are all champs with ridiculous burst and high mobility, making them great gankers.

Akali does not have to jungle to gank, She's effective in a lane as well. No other champion can burst over 350 damage at level 1 (takes some very good timing and positioning or else you'll only get half that damage in your burst, Mark-->wait-->Flash-->auto-attack-->mark-->ignite-->autoattack---> First Blood. Level 2 lets you get stealth through Twilight shroud, which makes it easier to pull off)

I've also gotten out of a number of sticky situations using her PBAoE stealth and flash by confusing the enemy to escape as well as else used her ultimate to jump to a nearby creep to avoid death.

Out of Annie, Akali, and Kassadin
Highest Burst: 1.) Annie, 2.) Akali, 3.) Kassadin
Best Escape: 1.) Kassadin 2.) Akali 3.) Annie (I personally think Akali might even be the best of the three, just becuase of her stealth and it's ability to avoid damage/break roots)
Best Chaser: 1.)Kassadin/Akali 3.) Annie


Ryze and Veigar also qualify as ranged burst. Run into Ryze 1v1 and your dead in no time flat (but then you have to wait for CD's) Veigars AoE Stun --> Meteor + Ult combo is instant death to AP casters (it's probably even overkill damage).

Annie is the best ranged burst till late game. When late game comes, veigar has more burst damage potential(only if he has been farming properly).

Kassadin