PsyBNC is a program you connect through using your IRC client to - usually - IRC with a different/vanity hostname. Example:
*!*DekuHaze@l33t.hostname.here.org
BitchX is an IRC client, like mIRC is. You use it to connect directly to an IRC network or to a bouncer (such as PsyBNC), and THEN to IRC.
Eggdrops are "bots". Bots are small programs that you configure to connect to IRC to try to look after "your" channel by setting bans on specified individuals, protecting against "floods"... etc. They serve as artifical ops, basically.
Hope that helps
