In EVE, the ship you are flying and the modules you have fit to it denote your role and base abilities in much the same way that character class and equipment would in a traditional RPG. Ships in EVE have a boggling number of statistics (from structure hit points to capacitor recharge rate to sensor strength) and every ship can be customized with rigs, drones and modules to be used in any number of different ways. That said every ship has a set of unique bonuses listed right in it's textual description which give a good idea of the primary role the ship was meant to fill.
Further, ships come in a variety of size classes, each of which has it's own broad role as well. You start off able to fly only the smallest ship class: frigates. In many way the frigate is equivalent to the "Thief" character archetype in many RPGs. Quick, hard to hit and able to do an astonishing amount of damage in the right circumstances.
As you progress, you will become better at flying frigates, and you will also learn how to fly larger and more varied ship types. It's important to realize though that bigger ships aren't necessarily better in this game. Being able to fly them just gives you more options. You can switch from "Thief" to "Warrior" (Battleship) to "Cleric" (Logistics Cruiser) just by docking and changing cockpits, once you've got the skills and fleet for it.
Of course some ships take longer to learn to fly than others, and some have narrow skill trees while others have incredibly robust ones. As you progress through the game, you will often find yourself having to decide whether you would rather learn to pilot an additional type of ship (and thus be able to, essentially, fill another character class "role") or continue to improve your abilities in one of the ship types you can already fly. One of the wonderful things about EVE's design is that this is almost always a genuine and difficult decision. Even after you are capable of flying battleships, frigates are surprisingly often the right tool for the job.
Rest assured that there are no shortage of ships to choose from. Combat ships are broken down into seven basic size categories. From smallest to largest, the combat ship classes in eve are: Frigate - Destroyer - Cruiser - Battlecruiser - Battleship - Capital - Supercapital
Each of the four races has multiple ships of every class. All of these ship classes except for Capital and Supercapital also have "Tech 2" and "Faction" variants which have more specialized roles, are more difficult to fly, are between 10 and 100 times more expensive and cannot be insured. And then there are the fantastically expensive swiss-army-knife Tech 3 ships.
Also there are plenty of non-combat ships in the game with hyper-specialized roles, mostly industrial. Although even these can be turned to non-traditional uses with surprising success. Witness the glory of the combat fit mining barge:
A note about how ship size affects combat:
Generally the smaller ships in the game are faster and better able to avoid taking damage, being particularly suited to hit and run tactics, while the larger ships are slower, harder hitting and better able to tank damage.
One on one combat among properly fit ships in EVE often comes down to a sort of complicated rock-paper-scissors. Say you have a frigate and are going 1v1 against a cruiser. In most cases, if you fly your ship properly, getting in close and orbiting at high speeds, you will win the fight because his cruiser sized weapons will be too slow to follow the movement of your fast moving ship. It will take a while for your smaller gun to chip through his shield and armor, but you'll have all the time in the world (unless his buddies show up), because he'll barely be able to hit you at all. Now, if the guy in the cruiser had known that he would be fighting a frigate, he could fit smaller faster guns and tear you apart, but that would leave him a sitting duck versus any other cruiser that was fit with regular cruiser sized guns.
This dynamic repeats itself as you go up through the ship classes. Of course, combat is much more complicated than that, especially between ships of the same class or when more than two people are involved and there are a lot of tactics that can be used to turn what should have been a loss into a win (or vice versa). Still, Sun Tzu's famous "The battle is won before it is ever fought" can be found oft paraphrased on the eve forums in the form of: "Most fights are won or lost at the fitting screen".
Let me leave you with an image (actually a link to an image, since it is far too large to display inline). This is a to-scale representation of most of the ships in EVE Online. About 95% of the ships in that image are player-pilotable. I'll freely confess that it was looking at this image for the first time that set the seed in my mind that this was a game I needed to play.
Here's the link: The Ships of EVE Online