In most fly fishing the reel isn't all that important. More often that not it's just a place to store your line. The exception to this is LARGE fish like big trout, carp, salmon or salt water fly fishing.
There are three basic types of fly reels, single action, multiplying, and automatic. The most popular type is single action.
Single action means that the retrieve rate is 1 to 1. Multiplying reels meant hat one turn of the handle turns the spool 2 or 3 times. They're not very popular because the spools are a pain to change and they're heavier than single action reels. Automatic reels are popular, with rod makers. 
Reason being that they put alot of stress suddenly on the tip of the rod and can cause them to break. They are not very popular with fly fishermen though. They are very heavy and the spool is extreemly hard or impossible to change.
Drags, there are two types of drags for fly reels, spring and pawl or disc. Both of these drag systems work well, so the choice of what to get really depends on what you fish for and where you fish. If you will primarily be chasing trout using light tackle, a spring-and-pawl fly reel drag system will likely be the best choice. Otherwise, a good disc-drag fly reel will work extremely well.
You don't have to spend a ton of money to get a decient reel. I use a Ross Cimmaron on my 3wt and I landed a 23" Rainbow on 5X tippet which is 5 pound test. But for most fly fishing a reel doesn't do much. You have to look at your situation and decide what you need.
fly fishing part 3, the reel
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