Hello wwpp,
Dr. John has explained very lucidly about the remote desktop connection. I am using LogMeIn as well when I travel.
There are different ways to connect via VPN. If you want very high security connection, Citrix has a VPN client that you install on the office desktop (or server) and your home (or laptop) computer.
VPN requires a static IP address for the office computer. You can request a static IP from the internet provider company like Comcast, Brighthouse, etc. Then you configure your VPN network connection on the computer. (How to configure the connection depends on what Operating System you have on the office computer like Windows server 2003 or 2008, XP, Vista or 7, etc. More or less the set up is the same except server OS like Windows 2003 or 2008). It is simple, a sample video for XP is here
http://www.youtube.com/watch?v=iNviKTWjU4c&feature=relatedHope this helps.
-Sachin