Your skills don't necessarily define what you can and can't do, but they do determine whether you're going to be successful at it or not. Skills range from 5 to 100 (perhaps higher or possibly lower through the use of some spell effects). Your seven Major skills start at 25 and your Minors at 5. Your Specialization (Combat, Magic or Stealth) adds another 10, although they may not necessarily be to Major skills. Your Race will also add points to some skills. None of the Birthsigns affect skills, although some will increase the controlling attributes. So a Redguard Warrior would see the Blade skill start at 45: 25 points for having it as a Major, +10 points for the Combat specialization, and +10 points for the racial bonus. The skill would probably start a few points higher, depending on how the Tutorial is played (you get to keep skill increases that occur during the Tutorial), so this character could conceivably start with a Blade skill of about 50. Not bad for a 1st level character.
The more you use a skill, the better you become at that activity, but you can also train a skill in order to increase it. One very noticable change in Oblivion is the restriction on the number of times that you can train skills: 5 times per level. That's not five times per skill; that's five times per level. It can be all in one skill or spread out over multiple skills. But after you have received five skill increases through training, you must gain a level before you can train again. Obviously, the higher your skill level, the better you are going to be at doing whatever it is. And, of course, the more you raise your Major skills, the more levels your character will gain.
Oblivion has implemented a rank structure with its skills:
Each rank has certain perks that get better as you improve in that skill. The perks are listed in the manual that came with the game, but I've also listed them on the appropriate skill page.
Magic skills have perhaps received the greatest attention in Oblivion. Unlike the chance of spell failure that you saw in Morrowind, your ability to cast a spell depends upon your rank with that skill. If you have the appropriate rank, you should be able to cast the spell successfully; if you don't have the appropriate rank, you can't cast the spell. The game conveniently tells you whether you can or not before you purchase a spell. When you purchase a spell, below the spell's effects you'll see one of three messages: