Static Difficulty:
This sort of system is pretty simple in concept--it's pretty much exactly what it sounds like. Each player goes through the game with a single, fixed, level of difficulty. It's surprisingly uncommon nowadays in comparison with other systems, although it used to be ubiquitous. Most NES games used static difficulty, for example.
Advantages: Every player gets the same experience. This means that the designer only has to make everything once, with no need for tweaking enemies, item placement, game mechanics, etc. depending on the difficulty level. This relative ease and focus of design can lead to gameplay that's better-balanced and more highly polished--at least in theory.
Disadvantages: Every player gets the same experience. Casual or crappy players can get frustrated, skilled players can get bored, and in general each player will have to conform to the game rather than getting the game to conform to them.
Selectable Difficulty:
The game lets the player set their difficulty. There are lots of little variations on this technique. Maybe the game lets the player control the difficulty of different components or sections of the game; maybe difficulty can be changed only at the beginning or maybe the player can alter it throughout the duration; maybe different settings involve a tweaked variable or two (like less health for the player) or maybe they mean entirely new level layouts, enemy placements, AI, etc.
This is a very common system today, particularly among action games like Ninja Gaiden II and Halo 3.

Advantages: Lots. Beyond the obvious benefit of tailoring the game experience to the capabilities of each player, different difficulty settings can be a great boost to replay value (particularly if there are lots of changes between each setting). Goldeneye on the N64 was one of the more prominent early examples of using difficulty as an incentive for replay; it did a lot of things right. Each level had individually selectable difficulty (with extra objectives and potential rewards for each setting), and that greatly multiplied the longevity of each bit of content. Modern games have taken that approach and run with it, with generally good results.
Disadvantages: Balancing one, static difficulty is hard enough--what about balancing, say, four or five separate settings? There's also the potential problem of the game only really working properly on certain difficulties. Halo sort of sucks on the easiest setting, and God Hand gets too frustrating to be fun on harder settings, to name two (arguable) examples of this problem. It's also not obvious to players which setting they should pick when they start, because the difficulty of each game isn't calibrated on some sort of universal scale. What's "easy" in one game may be considered "nightmare" in another.
Dynamic Difficulty:
This system entails the game automatically customizing itself around the player's skill level. SiN Episodes: Emergence (I felt the screenshot below was the best representation of the game) is an example of this sort of system. Basically, the better the player did, the more enemies would spawn and the more difficult and dangerous the game got.

Lots of other games have small elements that add some dynamicism to the difficulty. Remember the painkiller cabinets in Max Payne? The number of painkiller bottles in them was directly related to how much damage the player took.
Advantages: The problem of a player not knowing what difficulty to pick is solved with this system--the game will automatically adapt until it theoretically reaches a perfect level of hardness, completely matched to the player's skill level.
Disadvantages: Besides the practical difficulty of implementing one of these systems (shown by the many, many problems with the scaling in SiN Episodes and Oblivion; problems that made the difficulty in these games often veer wildly) there are some theoretical flaws. Making the gameworld revolve too heavily around the actions of the player makes it feel less real and more artificial and contrived. Besides obvious wackiness like lowly bandits wearing ancient magical elven armour because the player character is high-level, or saving the world as a lowly level 1 character, the whole enterprise starts to feel like a treadmill. The player simply cruises through, not making any real progress--after all, the better they do, the harder the game gets, wiping out their advantages
Organic Difficulty:
This isn't really a separate, distinct sort of difficulty system. Rather, it's a general approach and mindset to actually implementing a system. Put concisely, it's an approach where the designer integrates the game's difficulty in a natural-feeling manner rather than going with "gamey" contrivances like selectable settings.
So what does that actually mean? Some examples would help. Super Robot Taisen: OG for the Game Boy Advance is a game that does it (and does it very well). It's a strategy RPG with the usual basic structure common to the genre--you get party members, you customize them, and you fight a long series of detailed battles. But you don't pick a difficulty level at the beginning. The game also doesn't automatically adjust itself to your skill (at least, not completely). Instead, it uses a "skill point" system. Basically, each mission has two ways to complete it. You can either do the mission the easy way (by fulfilling basic requirements), or you can fulfill extra, harder objectives and complete the mission the hard way. For each mission completed the hard way, you get a skill point. Get enough of those skill points, and the game will present you with an alternate set of more difficult missions, with their own skill points to get. Keep getting skill points, and you can progress through the game with even harder missions. On the flip side, missing skill points will keep you doing easy missions.
It's ultimately an intuitive, flexible system that feels natural and meshes well with the story and gameworld. It's part static, part selectable (albeit indirectly), and part dynamic. I'd be hard-pressed to think of a better-designed difficulty system offhand, in fact.

Other examples occur in most RPGs (both Western and Japanese), in the form of optional content and sidequests. These sidequests can be of varying levels of difficulty. If the game gives some indication that a particular quest is harder than normal, the developer has essentially integrated organic difficulty into the game. The player can tackle hard challenges if they're up for them, or stick to the main, easier quest if they don't want a hard experience. Again, there's no need for difficulty selection, or any contrived or unnatural scaling system--just an organic, well-integrated difficulty system.
Advantages: Already covered; organic difficulty, when implemented properly, feels more natural than the other systems (which increases immersion), and can combine most of their advantages.
Disadvantages: Organic difficulty can be hard to design and can vary wildly depending on the game. It can also result in wasted content; if, say, 25% of the game is designed for expert players, that's 25% of the game that beginning/intermediate players won't see.
Again, these are just some of the major design styles for creating a game's difficulty. It's by no means an exhaustive list--there are probably other systems that don't really fall into any of these categories--but this list can serve as a brief overview and a way to spur further thinking on the topic of difficulty.