A VPN is an encrypted tunnel to one of your VPN provider's servers. You can think of it as a way to make all of your internet traffic seem as though it's coming from a different computer, but once it emerges onto the internet it's governed by the same rules that would apply if you weren't using a VPN. It's a good way to obscure the source of your traffic or to prevent someone on your network (like another person on an open WiFi network or your ISP) from seeing what you are doing.
First, no one should be able to see the files on your computer unless you have consciously decided to share them (or have been infected with malware).
The VPN provider can see your traffic, but if you are using a secure connection (https) your data is encrypted so they would be able to see what you're typing into an email or anything like that. If you aren't already, it's a good idea to use an email provider that uses SSL (https) by default. If you are using websites that don't use SSL, your VPN provider would be able to see your traffic.
As for "whoever is on the other end of that VPN server," it depends on what you mean. The website you're currently connected to needs to see the information you're sending to them (for example, if you're typing in your email password, your email provider needs to be able to receive that information). But they can't see details about your connections to other sites.
I'll mention one last thing. Not all VPN providers log traffic. Depending on what you're looking for, this may or may not matter to you, but here's a good article to help you find one that meets your needs: http://lifehacker.com/how-do-i-know-if-my-vpn-is-trustworthy-508866499