OpenRPG is a piece of open source software to enable people to play RPGs over the Internet. The code is written in the Python language with windowing done by the wxPython module, and so it is portable to any platform (operating system) that will run both of those. (This includes Windows and Linux, and eventually Mac.) You can download the most recent version from http://my.openrpg.com
The software includes a chat facility, with both private and public messaging, dice rolling (both public and to the GM only), miniatures mapping (with fog of war to allow for incremental revelation of the map), and storage of various bits and pieces of information in an XML format. Despite the complexity of all of these features, it's possible to only use a small subset of them and get a useful RPG session out. (Just using the battlemap, die rolling, and battlemap is 90% of the benefit, I think, and takes almost no effort to learn.)
The software uses a client/server model. You can set up your own server (which is relatively simple) and invite your players to join your server, or you can host a game on one of the many public servers available. (You can even open your own public server.) Often you can find new players to fill in the gaps in your own game by opening a public server and advertising.
Right now, the two competitors of OpenRPG are WebRPG and GRiP. There may be more, but those are the two that I know the most about. GRiP is closed source and commercial, but it has a relatively low pricetag. I've heard good reviews of the software, but I think that OpenRPG may have finally matched all of its features. GRiP does have a low-cost/free client available, so that the DM bears the cost of the software. WebRPG is free/low-cost, and also closed source.