As I pointed out in the Beginners Guide to Morrowind, "I can do something" is not the same as "I should do something." Because this is a free-form game, the things that you are capable of doing are only limited by the game engine and it does not put NPCs off-limits as the target of your baser desires. You honestly can kill almost anything in the game. Unlike Morrowind, however, the critical NPCs do have signs around their necks saying "I'm critical." Well, sort of. When you talk to them, you'll see a crown icon, which means that they are critical to at least one quest at some point. You can still whack 'em, but this is not what I would classify as a "terribly bright thing to do." At least before they stop being critical.
Each NPC that you kill (if you are observed, anyway) puts a bounty on your head. In Morrowind, they were nuisances that didn't really interfere with anything unless you tried to talk to a guard or unless you racked up more than 1000 gold in bounties. Not so in Oblivion.
For the most part, bounties are piddly little things that you pick up for doing something odd-ball (assault, theft and murder being notable exceptions). For example, I thought it would be fun to move one of the books in the bookcase next to Jauffre at Weynon Priory. Jauffre went ballistic, so I stopped (yo! get a grip, dude). He calmed down again, we conducted our business and I went on my merry way. I didn't know it at the time, but I had picked up a 4-gold bounty for that little escapade (probably attempted theft or trespassing, because that would have been the approximate value of the book I was moving). I didn't know anything about it until I got decently spanked at Kvatch and found that couldn't heal myself in the Chapel ("Repent, you sinner" or something to that effect - and wouldn't you know that there was no guard around where I could pay the fine). Sometimes bounties will also prevent NPCs from making offers that they would normally make. Jauffre, for example, did not offer to let me join the Blades because of that bounty.
You also have an additional statistic that the game keeps track of: Infamy. Fame is now a point score of the good things that you have done. People tend to like you more when you have a high Fame score and there are some things in the game that cannot be done with a low Fame score. On the flip side of that equation, your Infamy is a point score of the shadier things you have done. For example, just joining the Thieves Guild will bump your Infamy by one pont. In order to get healing at a Chapel, your fame must be higher than your infamy.
OK, so all of that out of the way, here's the breakdown on crime in Cyrodiil. Whatever it is that you do, you have to meet one condition to wind up with a bounty: you have to be observed committing the crime. This means that there has to be someone around that can see you do the crime and is capable of reporting it (you can commit all of the crimes you want in the presence of barnyard animals). If there is a guard within earshot, you will probably be arrested and offered the chance to pay your fine, go to jail or resist arrest. Resisting arrest is not recommended. The guards in Oblivion are scaled NPCs and should always be 10 levels higher than you. Going to jail is also not recommended as your skills will randomly decrease at the rate of about 1 point per day. There is a slight chance of your Security and/or Sneak skills increasing, but you can actually make that happen by just sneaking back and forth in your cell until you get tired of doing it. And since you always get to keep one of your lockpicks (this is one issue where we don't want to inquire too deeply - eewww!), you can play with the lock on your cell until the pick breaks. So I'm going to assume that you will choose to pay the fine.
Trespassing (bounty: 5 septims): you go where you shouldn'ta oughta went. You can tell if you're tresspassing because the door icon will show red before you enter. Going in is not always, in and of itself, trespassing. In some areas you do have a short (30 seconds to a minute or so) period in which you can get in, do your business and leave without raising an alarm or incurring a bounty. But if you hear an NPC say something about you're not supposed to be here, that's a real good clue that you're trespassing. In a few areas, just being detected in the restricted area is enough to send out an alarm. But if you say what you need to say and then get out, most of the time it will be forgiven.
Theft (bounty: the value of the item): taking what isn't yours. Sounds simple, but it's not. The key is the status of the item you're taking. If you see a red hand icon before you pick up something or if the container shows red before you open it, then you're stealing. Items that belong to people who are part of evil-aligned factions are generally free for the taking and, except for the Arena, feel free to loot dead bodies. Also, short of a modding (and there are at least a couple out there), regular merchants will not buy stolen items. Unless you're planning on using it yourself or selling it to one of the Thieves Guild fences, stolen merchandise just takes up space.
Pickpocketing (bounty: the value of the item; there is a minimum fine of about 20 or 25 septims): it works just like theft except that you're swiping something that is already in someone's inventory. You must be in Sneak mode in order to pickpocket and you'll see a red-hand icon when you try to do it. It's quite easy to accidentally pickpocket and it happens when you try to initiate a conversation while in stealth mode. The fine applies even if you don't take anything. I accidentally pickpocketed one of the Guard Captains in the Imperial City by trying to talk to him while I was in stealth mode. Ooops!
Assault (bounty: 40 septims - Guide says per level, but I believe it's a flat fine): smacking someone belonging to a non-evil faction. It's OK to smack people belonging to evil factions; just ask Minsc. NPCs already engaged in combat might forgive an errant shot or two as being "friendly fire," but you'd better believe that non-combatant NPCs will not tolerate it. Depending on how seriously they feel threatened, they'll either attack you back or run away. If you can take 'em down in one shot, you might be able to get away with it. And I don't believe that there is a penalty for using spells.
Murder (bounty: 1000 septims): you just up and kill someone (always an NPC - creatures don't count). You are permitted to defend yourself when you are attacked, though. Unlike Morrowind, where you could Taunt an NPC into attacking you, you're going to have to get creative in Oblivion. And remember those "critical" NPCs that I mentioned earlier? Some of them can't be killed under certain conditions or until certain quest stages have been reached. I mean that literally - they can't reach 0 health points. Well, they do, but they just kind of fall down for a bit and then get up again. If there is no one around and if you can take down your target in one blow, you might be able to get away with an unprovoked killing (from the guard's standpoint - the Dark Brotherhood's "unseen eyes" is another issue entirely). One trick that sometimes works is to reduce the target's disposition to zero and then cast a "Frenzy" spell on them. Under some conditions, this will cause the NPC to go hostile and attack you, but it's a hit-or-miss proposition in my experience.