FMC stands for fewest moves challenge. It is hosted by Dan Harris at his website. There are several approaches to this and I am just going to show the main ones I try when I enter the contest. I would like to point out that I'm not very good at FMC, but I usually only spend a couple of hours on each one, and hopefully with practice I will get better.
    If you really want to get good at FMC I would suggest visiting the . There is a ton of information in the archive. Just search for help or ask questions.

2-Generator group - The best page for this is Charles Tsai's page which is currently not up anymore. Very unfortunate. Basically you make a 2x2x3 block and then orient all the edges while placing the corners and then finish the F2L, and usually you can find a somewhat simple LL case to orient the corners and position your edge pieces.
    I used this method for two of the contests, but I could never get good LL algorithms and I didn't do so well. I don't think I'll use this method hardly ever, but it is very good to know it, and I recommend trying it at least a couple of time.

Insertions - This is the most commonly used method for good FMC solutions. Basically you solve as much of the cube as possible and leave 3 or 5 corners unsolved. Then go back and insert an algorithm somewhere in your solution to fix the corners so that they will be correct when you get there. You can use insertions to fix edge pieces as well or a combination. The best description for solving corners is at Ryan Heise's webpage. He has a wonderful tutorial.
    A real advantage to insertions is that you can usually find a place to insert them where they will cancel a move in your solution and give you a lower number of moves!

The rest of these are approaches that I use and not really methods. All of them use insertions to finish out the corners. I'll go through my thought process.

Approach 1
It would be extremely helpful if you knew the Petrus method. This method is kind of the springboard to all of these approaches.

Step 1 - Find the shortest 2x2x3. If I don't find one within 8 or 9 moves I usually scrap it. This step can be really aggrivating, but is also the one where you can use the most imagination.

Step 2 - Check 2 generator group. Like I said I usually don't use this method but I still glance to see if an easy solution is there.

Step 2 - Try to solve all but three corners. If I can't do this in 26 moves I move on. This can be a really hard way to do it while the next is pretty easy, but occasionally something works out really nice.

Step 2 - Solve all of the edges and one corner. This is much quicker and easier, but also leaves 5 corners unsolved which requires two insertions. I move on if I can't do this in 20 moves.

Step 2 - If there are a several bad edges I check solving it to a state that M U M' would fix provided your 2x2x3 is opposite the front.

Step 3 - Find a good place in solution and place insertions.

Approach 2
These are based on the Roux method.

Step 1 - Find the shortest 1x2x3 you can. This should be doable within 5 or 6 moves.

Step 2 - Build the other 1x2x3 block or build them simultaneously. Then fix edges which I usually do while building the block, and then place the edges so everything is solved but 3 corners.

Step 2 - Place the 3 edges for the other block. This can be done in just 4 or 5 moves, plus you can orient the edges while doing this. And you can place them while building the first block. Then you place the rest of the edges and are left with 5 corners unsolved.

Step 3 - Find a good place in solution and place insertions.

There really are an infinite number of tricks and several excellent explanations are(were) at Charles's page.

Pseudo-blocks - This is just using pieces for a block that can be easily moved into place later. Like solving the entire cube to the R' position and at the end just doing R. I'm not really explaining this at all, but Ryan and Charles both have really good explanations.

Inverse solving - This was shown on Charles's page and is one of the coolest tricks I've seen. You basically take your solution to a given section and apply it to a solved cube, then try and find a shorter solution. Then the inverse of that will be a shorter solution to the original scramble.

My FMC Solutions
Later. For now just search my name at the site.