Fly reel
A fly reel is a centrepin too: the drum sits directly on the axle and there is no gearing. But its defining feature is not that — it is that it takes no part in the cast.
Your hand casts, not the reel
In every other method the lure pulls line off the reel. In fly fishing it is the other way round: what flies is the heavy line rather than the fly, and the angler pulls the length he needs off the reel beforehand and holds it in loops in his hand or on the water.
So the reel here is a store. It comes into play only once the fish is on and running, taking up what was lying in those loops.
Hence what is asked of it
Since the reel does not cast, it needs neither gearing nor a fast retrieve. It needs three things: to hold the store, to give it out smoothly, and to balance the rod — a fly rod is held out at arm's length longer than any other, and weight at the butt is felt directly.
The drag matters more here than anywhere
A fly tippet is fine and the fish often runs against the current. The drag has to give from the very first millimetre — a jolt at the start on such a tippet means a break. It is the same fault described in the entry on the drag, only the price of it is higher.
On simple reels the brake is a ratchet check — a sprung pawl against a toothed wheel. On more elaborate ones there is a disc drag, built the same way as on a spinning reel.
Why the arbor is large
A drum of large diameter — a «large arbor», literally a large hub — takes up more line for the same turn. That is exactly the point made in the entry on gear ratio: where there is no gearing, the metres come from geometry.
The second gain is that line bends less sharply round a large diameter and keeps fewer coils from sitting on the reel.
Backing is not optional
A fly line is short, and on its own it does not fill the drum. Backing goes underneath it: it fills the drum and also serves as reserve when a fish takes all the fly line. The procedure is the same as in ordinary spooling.
A handle that turns with the drum
On most fly reels the handle is fixed to the drum: when a fish takes line, the handle spins as well. That takes getting used to — the hand comes away at that moment rather than trying to hold it.
There are designs in which the drum turns without the handle on the outrun. That is more comfortable, but it is one more assembly, and an extra assembly in a reel that spends its life getting wet is not always a gain.