What is a fixie?
A fixie, or fixed wheel bike, has a single cog fixed to the wheel, so it will spin forward or backwards with the cranks, but can't spin freely. You can put back pressure on the pedals to slow down, or lock your legs to skid to a stop.
Riding a fixie can feel more connected with the bike, since you're always in motion with it. You will find yourself speeding up your approach to hills and generally riding more efficiently.
Fixed gear hubs
A modern bike wheel has a freehub, where the cassette attaches. It has a ratchet system that allows it to freely spin backwards, but locks to rotate the wheel when spun forward.
A fixed gear hub has threads for a single cog, and reverse threads for a lock ring. This is so that when you backpedal, the force will tighten the lockring instead of loosening the cog.
Look for track hubs, which have the right threads for a cog and lock ring.
Flip flop hubs
Some wheels have a fixed gear on one side and a freewheel on the other to let you flip the wheel around and switch between the two.
Fixed gear frames
You can go with a track specific frame that has horizontal dropouts.
You can also use a vintage frame that has diagnal dropouts.
If you really want to use a normal frame with vertical dropouts, you can find special hubs that can adjust forward and back in place.