Ah, biketrials.
I spent about 5 years learning how to ride trials. I never got that good, though, and ended up with about a dozen stitches for the efforts.
"Where's the seat" depends on the bike. If the bike wasn't specifically designed for trials, then usually the seat's just dropped really low. Yes, so you don't catch your nards. There are seats specifically made for trials, I believe they're called "uni" seats. These seats are really, really tiny, and are there more to give you a little more protection from ramming a seatpost up your exit hole, than for sitting on. Again, it's just to get it out of the way. Some of the trials specific bikes (the Monty brand, in particular) design the frame with nowhere to put the seat anyway -- the seat tube is just covered over.
I haven't watched the videos (yet), but, just so you know...
There are two types of bikes used for observed trials (OT) -- stock, and mod. The stock bike looks mostly like a regular mountain bike, though maybe a bit small for the rider. The mod bikes look kinda like a goofy BMX bike, with big fat rear tire, a tall skinny stem, and not much of a chainring at all, protected on the bottom by a bash plate.
For more of the same type of stuff:
Ryan Leech: one of the best trials riders in the world. The guy is exceptionally smooth, has the same birthday as me, and is generally an all-round nice guy.
Biketrials.com: Plenty o' resources.
Hans Rey: Probably one of the most famous trials riders, due to a) his talent, and b) his videos (which are what got me started).
Kris Holm: You think those videos are amazing, check out Kris. He rides a unicycle. Un-frickin'-believable.
Dang. You made me want to get a trials bike, again.
Hmm...