pretty much what OOO has said, the tuck is the get the bike higher without it hitting you in the arse and the twist is to shorten the time the bike takes to get over the bar.
Which way you tuck is personal preference, Danny, Myself and most BMXers will tuck to our back foot side as this is the same way we spin so it means you can do high 180 hops and general spins in riding (think tyre tap over things).
Trials riders tend to tuck to the other side as you can tuck the bike up higher in sidehops and not be limited by your back foot being in the way. The down side is that it can really mess with how you bunnyhop and link everything together when riding street. I had to unlearn my spins and tucks and start again the other way, I really think it helps in the long run.