A VPS is a virtual private server -- roughly equivalent to a shared hosting account, but emulating a dedicated server with root access (or equivalent) instead of just a Web site. Typically these offers come with significant "fair use" restrictions to make sure no one customer totally overloads or monopolizes the physical resources, starving the other VPS's hosted on that machine.

I've actually been having the opposite problem from oldbushie.  On my VPS, I've installed a monitoring utility (monit).  For some reason, monit determines that p2pool and bitcoind aren't running (the email explicitly says the trigger is that the process isn't running, not that one of the other trigger conditions is met), about once an hour, and promptly proceeds to kill and restart the existing sessions.  This is surprising, since both ran just fine for days before I installed monit.  My guess is I configured monit wrong.  If anyone has experience debugging monit scripts or settings, I'd appreciate any feedback.