Your character's class pretty much defines your character in terms of gameplay. Oblivion continues Bethesda's tradition of keeping games as open-ended as possible. Aside from, or perhaps including the standard or "canned" character classes, you are perfectly free to develop your character along whatever lines you'd like. Your character's class is defined by the skills that you choose during the character generation process, but the armor and weapon restrictions that one normally associates with role-playing games (i.e. Mages aren't allowed to wear armor), aren't present in Oblivion. Any character can use any weapon, wear any armor or use any magical item or spell that they'd like, but some classes are going to be better at it than others. Mages, for example, are going to be more adept at casting spells than they are going to be at swinging a Battle-Axe. Warriors are going to be better at swinging Battle-Axes than they are at picking locks. But if you'd like your character to be a Battle-Axe wielding, fireball-slinging lock-picker, you can, even with one of the standard classes.
Personally, I tend to let the standard classes fall by the wayside (though I must admit that Nightblades are way cool). Inevitably, though, the standard classes will probably use skill slots for things that you don't need or want, so feel perfectly free to abandon them and create your own custom class (that's what the big "Custom Class" button is for). Note that the name of your Custom Class must be 10 characters or less and will be added to the list of available classes in case you'd like to choose it again. Novice players might want to stick with one of the standard classes (pure warrior or thief, for example) until they get a feel for the game. But this is very much a "do what you want to do and go where you want to go" kind of game, so don't feel constrained or restricted by any recommendation that you see here.
The 27 skills of Morrowind have been reduced to 21 in Oblivion, but they are otherwise pretty much the same. Character skills are divided into three areas of specialization: Spells (magic), Steel (combat) and Sneaking (stealth). If you choose to make a custom class, you must also choose a specialization. Skills in your specialization receive a bonus and are easier to develop.
If you decide on a custom class, you must also choose two Favored Attributes. These two attributes will receive a +5 bonus at the start of the game. The important part is that each skill is governed by one of your attributes, so the higher the attribute, the better you are going to be at doing that skill. Note that Luck, while having an effect on just about everything you do, does not govern any of your skills. This means that you will not be able to see a multiplier with this attribute, so it can only be raised one point at a time.
For Oblivion, the multi-tiered approach to skills (Primary/Major/Minor/Miscellaneous á la Daggerfall, or Major/Minor/Miscellaneous á la Morrowind) has been revamped into a two-tiered approach (Major/Minor). You have seven Major skill slots to fill and everything else gets dumped into Minor skills. This is not necessarily a bad thing, but be aware that Major skills receive starting bonuses and will be easier to develop than Minor skills.