The automatic fly reel is not used much today; its use peaked in the '60s. You may not see them among fly fishing purists, but they do have a place.
The automatic reel is spring-driven. The spring is wound when the fly line is pulled out and recovered by pressing a trigger. The spring is released by a knob on the side of the reel. It is important to release the tension on the spring after you rig your line. (See the section below on fishing with an automatic reel.) It does not have a handle or a crank.
The automatic fly fishing reels you will see most often today are the Martin and the Pflueger. They have a historic status similar to a wicker creel. You may be able to find a nice one cheaper than what you will pay for a fly line. You may even be able to find one at a garage sale.
Each double sided box holds flies that represent different stages of a insects life cycle for either Mayflies, Caddis flies or Stoneflies and includes a laminated card listing the flies and their hook size so you can restock the box.
Since these reels will not hold a lot of backing you only need about 50 feet of 30# test Dacron backing. I would use a weight forward floating fly line.
These reels can bring in line very fast so be careful. When setting up your rig, be sure to release the spring when you have about a rod length of line out past your rod tip. The reel becomes "charged" when you pull line out and if you don't release the spring you could accidentally bring your fly back into the guides. You may break your rod tip.
After you have released the spring, you can then start to pull out more line, which will wind up the spring again. You can be confident at this point that the reel will not pull the line in any further than where it was when the spring was released.
You do not use the reel when fighting a trout. This is one reason why you don't have a drag system on an automatic fly reel. These reels are used only to store line. During the fight the angler should strip line in by hand and if necessary pull the trigger to store slack line. This is very helpful if you are fly fishing from a boat or in weedy and brushy areas.
Unless you have mobility issues with your hands I would not recommend an automatic fly reel for your first reel. (Save it for your fourth or fifth, but only if you can find one at a good price.)
You take all the fun out of fly fishing when you remove all that line at your feet. Nothing is more fun than stripping in line as fast as you can while a large trout has decided to run in your direction. When it then turns and makes a run for the next county, the tangles at your feet can be fun.
The whole experience builds character. You need more character, right?
Start with a single-action fly reel. Keep it simple!