Vice
The vice is by far the most important tool in fly tying. There are clamp-mounted and base-mounted vices. Clamp vices are more secure; although, base vices are very nice for traveling.

Scissors
The scissors are very important. Try to get scissor made just for Fly Tying. I personally like micro serrated edges.

Bobbin
The bobbin holds the thread and on sometimes other materials(floss). If you don't have a bobbin just forget about tying flies. You only need one of them. Ceramic tubes are teh best but are not manditory.

Other Tools
These tools will often come in handy: pliers, bodkin, blender, dubbing twister, old scissors, material holder, whip finisher, half hitch tool, wing burner, bobbin threader, hair packer, hair stacker, hackled gauge.

Here are some very helpful animations:
(press refresh to view animation again)

WHIP FINISHER STARTING THE THREAD animations by Derek Porter