You need a rod, a reel, and line, just like any other kind of fishing. But
a fly rod bends in a specific way, so that you can put the fly where you
want it, but also to play a fish without breaking your leader.
For much of fly fishing, the reel is simply a device to store line, and you
retrieve and otherwise manipulate the line with your hands.
Sometimes, when a big fish is hooked, the reel takes over to provide a
mechanical drag and to retrieve line between runs.
A fly line has weight, because the line is what casts the fly. A fly line
can float or sink, but for now we'll stick with a floating line, which is
by far the most common type.
Between the line and fly is a leader, made from regular monofilament
fishing line that has been specially tapered to present the fly properly.