Most massively multiplayer online (MMO) games handle skills, abilities, and leveling the same way. You create your character by choosing your race, which sets some stats, then you choose your class, which may modify those stats further and add some features, and maybe you have some points to sprinkle on your stats and early skills before you drop into the game world to start a tutorial.
You level up by earning experience points via combat and quests, and gaining levels increments some of your stats, often based on your class and even race. These stats control your effectiveness in the game. Stronger players swing clubs better, smarter players fling magic better, and faster players dodge better. Leveling can also give a point or two to spend on special abilities or skills every so often.
Where to spend these points can be a difficult decision, but as time goes on player guides appear telling you the "right way" to spend points to get the most bang for your hard-earned skill point bucks. Skills in one area often keep you from trying skills in other areas without starting over, and your character joins a larger population of me-too players who have been configured to follow the "best" way of doing things. It behooves you to follow these guides since you're paying to play the game and want to get the most value and least frustration for your money. And if you discover these guides after you start assigning skills you think would be useful? Well, you can always start over.
EVE Online differs greatly from all of those other games. There are no levels, and your character's stats do not affect how you play the game. There aren't any experience points in the game, either, and your skills actually do not improve with use. To be fair, it does track your standing with various factions, and performing favors -- combat or otherwise -- for them does get recorded, but you don't receive the universal reward of experience points like you do in other MMO games. From the classical MMO standpoint this sounds almost heretical and at first glance doesn't seem to make sense, but a quick run through of how the skill system works should clarify things.