Here's a quick overview of the main classes in Mass Effect 2 that you'll see when you first start out with the game.
1. Vanguard
The vanguard class is known for its high risk, high reward style of fighting. With the vanguard you should expect to be getting up close and personal with your enemies on a regular basis. Biotics are used with this class.
Your main ability will be charge, after which you can follow with melee strikes or weapon blasts. You have a variety of weapons to choose from including pistols, shotguns, and assault rifles.
2. Soldier
The soldier class is fairly straightforward: you get a large amount of weapons, better ammo choices, and higher sustainability. Adrenaline rush is an ability you'll receive, allowing you to slow down time and take careful actions against your enemies.
The drawback of the solider is the lack of biotics; if you don't care about biotics, this class is a good choice for you - or if you're just starting out in Mass Effect 2.
3. Infiltrator
The infiltrator class is the sniper class of ME 2. You'll get an ability which lets you sneak around undetected so you can get the best shots at your enemy. If you like playing a sniper in your games, this is the class you'll want to pick.
Drawbacks of the sniper: again, biotics are limited. As well, you'll die pretty quickly if you get hit out in the open. And it's hard to avoid being out in the open in a lot of situations. You'll need to get good at directing your team to take care of thickly-combatted situations to be really good with the infiltrator class.
4. Sentinel
The sentinel is a biotics based class in ME 2. This one is great if you don't like a whole lot of shooting and would prefer to be a ranged "magic" user. Well, biotics in place of magic, of course.
5. Engineer
The engineer is a class that can reprogram synthetics and use them to attack enemies, as well as several other cool machine-based abilities. This class doesn't use a whole lot of biotic abilities, but in their place, you have abilities with machines and instruments.
You do also have access to certain select weapons like the pistol. You'll need to carefully allot your skill points so you can reprogram enemies and overload shields and systems to destroy them.
These are some of the main classes in Mass Effect 2 that you'll want to check out when you start playing the game. Each one has unique challenges and highlights, so spend some time playing around with each one til you find the one you like best.