It stands for Representational State Transfer and it can mean a lot of things, but usually when you are talking about APIs and applications, you are talking about REST as a way to do web services or get programs to talk over the web.
REST is basically a way of communicating between systems and does much of what SOAP RPC was designed to do, but while SOAP generally makes a connection, authenticates and then does stuff over that connection, REST works pretty much the same way that that the web works. You have a URL and when you request that URL you get something back. This is where things start getting confusing because people describe the web as a the largest REST application and while this is technically correct it doesn't really help explain what it is.
In a nutshell, REST allows you to get two applications talking over the Internet using tools that are similar to what a web browser uses. This is much simpler than SOAP and a lot of what REST does is says, "Hey, things don't have to be so complex."
Worth reading:
- How I Explained REST to My Wife (now available here)
- Architectural Styles and the Design of Network-based Software Architectures
