You're not alone. Whenever you visit a website, the server records information about you in something called a log file. The site's publishers then use log-analyzer programs to harvest this information in a variety of formats. You can think about an IP address like an address in the real world. For the most part, everybody has one, and it represents the location of their machine on the Internet.
Visits are the total number of sessions on a site over a given period of time.
A session starts when you arrive at a site and ends when you leave it. Regardless of the number of pages you look at while you're on a site, it's still only one session. However, if someone were to visit and leave a site in the morning, and then return a few hours later, that person would appear as two visits for that day. These are, very simply, the number of pages a person sees on a site. Say you visit Yahoo and click around on ten different pages.