I hae a big ip 2000 appliance for load balancing, tell me what it can do!

By default does it proxy the connections?

if you use tcp profile, yes.

sol8082: Overview of TCP connection setup for BIG-IP LTM virtual server types

Can it send mail alerts ?

yes

sol3667: Configuring alerts to send email notifications

If say the web service goes down on one of te severs, could it restart it? what intelligent things can they do ?

i do not know. it may be possible if server provides interface to bigip to restart it.

Acton on Log - using the alertd deamon

What is persistence ?

do yo have any specific question?

Manual Chapter: Session Persistence Profiles