How do enemies and monsters level up?
It works like this: Certain things in the game world scale with your character linearly, like they did in Oblivion, and certain things are "semi-static" and "semi-scaling", meaning that they scale to a certain extent, but the scaling has limits.
The latter one consists mostly of dungeons. All of the various dungeons in the World have a static, pre-determined difficulty which is described in terms of a range of the players level.
For example, there are a lot of dungeons early on which have the difficulty level "1-10", and that's where they'll always be, they won't scale beyond that, and similarly, dungeons of higher difficulty level will not be manageable by you until you level up some more.
This is also the reason why the game still feels very leveled despite having a different scaling system. A lot of the dungeons and areas that you encounter around the part where you start (southwest near the mountains) and north towards and around Whiterun are in this level 1-10 area, only slowly progressing more upwards around the edges of that area. For example, the first dungeon with a higher difficulty which can be reached with ease is, for example, a dwemer (practically the "tinkering dwarfen" race of the Elder Scrolls games) ruin that is accessed via two levels of caves just to the north-east of Whiterun, embedded into a small mountain. And once you start visiting the other places, you will usually have leveled beyond level 10 and the game world will scale fairly well with you.
An exception to that is the case of "cleared" dungeons and possibly areas. When you have previously cleared a dungeon, it will say so on the map when you hover the cursor over the location's legend. These cleared dungeons will eventually get repopulated after a long time (it's longer than just a couple of days, but I don't know the exact number), and when they do, they repopulate with mobs in a level-range that corresponds to the players level.
The actual level of the monsters inside those dungeons is determined by your first entry into the dungeon. When you first enter the dungeon, the level of all monsters inside it is "locked" to your level. For example, if you enter a dungeon of difficulty "10-20" with level 14, all enemies will be set to level 14 and will be locked there, so that if you find it too hard when you first enter it, you can come back later and try again with level 18, at which point they will still be level 14.
The difficulties in that case are clamped, which means that if you enter a 10-20 dungeon with level 5, all enemies will be set to level 10, and if you enter it with level 25, all enemies will be set to 20.
Also, the areas in Skyrim work roughly that way, too. In general, you will find more difficult monsters the higher you get in the world (near the mountains). An obvious example that comes to mind are the frost trolls and sabre tooth cats, which will beat you up badly, if you walk around in the mountains too much early on. But as opposed to dungeons, monsters won't be locked into a level once you enter the area. Instead, the game randomly sometimes spawns monsters that are a couple of levels below or above you, or equal to your level.
One of the things which has "linear scaling", almost exactly like in Oblivion, should be dragons. Todd Howard has said a couple of times that he wants every player (no matter how good or bad) to be able to experience at least all of the main storyline. Although even with those, it will probably be an advantage to have better equipment or perks which give you some kind of resistance to something (for example, there are frost and fire-dragons, and a couple of other types).
An example of monsters who are probably completely static in level are giants and mammoths. Ever fought a giant with Level 1?
Notice that this only concerns the level scaling of monsters and other enemies around the world of Skyrim. As pointed out by SLC in the comments, items seem to fully disregard the semi-static level scaling of dungeons and simply scale linearly with you:
"Incidentally, although the noob cave at the start with a quest to kill some bandits is laughably easy later on due to them not scaling, the loot DOES scale. I went and did that first kill the bandit leader quest, which involved easily killing about 3 bandits, and got an ebony bow and some other high level items." – SLC
In addition, there are scaled items, which are usually quest rewards and depend on the players level at the time of quest completion, similar to how dungeons scale.
Also, this might count as kind of a second-hand source, but at least it's an interesting read. It's a little explanation of how the scaling worked in Fallout 3. If you Google for "Skyrim scaling" you will find a lot of references to devs saying that the level scaling is somewhat similar to that of Fallout 3.
Enemy level generation, from UESP wiki, "Effects of Leveling":
...all leveled enemies are generated more like leveled creatures in Fallout. For example, Bandit NPCs are always a fixed level for their name (Bandits are level 1, Bandit Thugs are level 9, Bandit Highwaymen are level 14, etc). The player's level affects the range of possible bandit types generated within a bandit dungeon, and probably the frequency, but does not seem to affect the resulting stats except in a few rare cases. Lower variant bandits remain reasonably common even when more dangerous bandits are available.
Well, more or less, but then again not really. In my prerelease excitement of the release this Friday, I spent an evening searching for interviews and coverage on Skyrim on youtube, some actual footage on streaming sites of people who're already playing the game on the XBox, as well as for articles on the internet. I watched and read a lot, so it will be hard to find the precise sources of these things. I realize that giving sources is one of the most important things of all, but I figured better answer without sources than no answer. But I'll try finding the things again, starting right now.
added a source about fallout 3 scaling, just for reference. Also, I'm going to edit the answer just slightly to reflect what I learned about dungeons repopulating.
Incidentally, although the noob cave at the start with a quest to kill some bandits is laughably easy later on due to them not scaling, the loot DOES scale. I went and did that first kill the bandit leader quest, which involved easily killing about 3 bandits, and got an ebony bow and some other high level items.