Ahh - the question that has plagued raid leaders for years, or at least since 10m were introduced in BC. Raid composition and roster juggling is one of the major headaches for any raid leader or GM. Maintaining a bench is hard enough, but making sure your bench isn't just filler is even tougher. Don't rotate often enough, and your bench players won't have the gear or experience to get you though those new nights. Rotate too much, and nobody gets enough. It's a fine line to walk.
Tanks, healers, and special cases
Regardless of what you might want, there are certain players who are always needed - at least one tank, two healers, and any special cases (legendary fragment recipients come to mind) that must be in a raid. Now, a good raid leader will make sure that every tank/healer combo is a) valid and b) has time on boss. It doesn't do you any good when your 3rd tank has to become the MT for a fight, and they've never seen it before. The same goes for anyone in a 'clutch' role. If only two people know how to do Magmaw's chains, and one is missing, you're probably going to wipe the first time someone new steps in.
In a standard setup though, you're probably going to need at least two tanks and two healers. Adjustments after that are generally case by case.
How many healers is enough?
Most guides I read call for a two tank, three healer, combo for every fight. This setup is pretty safe, since it gives you extra healing for those AoE moments, and gives the healers a chance to compensate for poor play on the part of the DPS and tanks. Now by poor play, I mean the combination of several different things. Tanks that don't use CDs, players that take avoidable damage (flame waves on Ragnaros are NOT delicious, nor are they fun to ride), and other healers who have poor spell/mana control. The other thing a third healer gives you is flexibility of playstyle.
Flexibility of playstyle isn't compensation for poor play, but instead is a different mindest to work off of. With a two healer setup, each healer has to be completely fluid in their play. You can't go with static assignments on the majority of fights (Chimaeron is an exception to this), and you have to be able to cover a lot of players in a hurry. This type of healing really requires your raid to avoid as much damage as possible, and to use self healing where applicable. This type of healing is (in my opinion) a lot more fun (read - stressful) since you can often find yourself on an island without backup.
Why three healing is better than two
Three healing fights instead of two healing gives you one major advantage - healers can focus on specific assignments. Raid healers can ignore the tank, Paladins can do what we do best - keep the tank(s) alive, and DPS can use fewer CDs on self healing (sloppy play will still get you dead). This isn't my preferred style, but there's nothing wrong with doing it. It also allows you to have a healer absent, and still work through two healing an encounter - solo healing is going to be pretty tough in most of these encounters.
Why two healing is better than three
Things die faster. Bottom line. Let's say your DPS is all sitting within 3k of each other - so you have a range from 17k to 20k on any given boss fight (yes I know numbers fluctuate all the time, blah blah blah). Now, if a boss has say 67M HP, and your raid is doing 100K DPS - it's going to take you 11.1 minutes to defeat that boss. If instead, you bring 120K DPS, you're now looking at 9.3 minutes. That's pretty significant. Bosses dying faster means you're though transitions quicker, healers need less mana, tanks have less "scary" time in a certain phase - everything is just faster.
The bottom line
The real bottom line though is that you have to do what's right for your group/guild, and for the specific boss. For example we 2 healed H-Atramedes, but had to use 3 for H-Chimaeron. Even though I totally believe we 'could' have done H-Chim with two, there were just too many insta-gib moments to make it worth it. Even with three healers we'd still lose a tank to a missed block, a slow 10k top off, etc. H-Atramedes and H-Maloriak on the other hand just required that the raid avoid all the incidental damage possible.
Your raid makeup also plays a big part in this. Getting a lot of incidental healing from a healing stream totem and vampiric embrace? Are your raiders using GCDs to heal themselves with things like recuperate, word of glory, or drain life? Are your tanks avoidance or stamina tanks? Are your healers even comfortable two healing? Does your Warlock life tap right after you've topped off a raid and told the mage to go ahead and trigger Ragnaros's trap, sending someones blood pressure straight to the top?
All of these things, and more, play into what kind of healing strategy you bring to the table. The only wrong way is the one that doesn't get bosses killed. As long as the bosses are dying, use five healers and three tanks (in 10m) if you need to. You may not get the fastest kill in the world, but if you're having fun doing it, who gives a shit.