OpenSimulator (often called OpenSim) is an open source platform for creating virtual worlds.
OpenSim was originally inspired by Second Life. Second Life is very popular among virtual world enthusiasts due to the great degree of freedom it gives its members, by allowing all users to create their own objects and scripts. The greatest limit to this freedom is that the entire world is owned and governed by a single company, Linden Lab, whose laws and policies are often extremely unpopular (the complaining is constant in some forums). OpenSim is the solution to that limit.
In 2007, the client for Second Life became open source, allowing programmers to create their own programs to connect to Second Life. Rumor was that they would also open source the server program, but at that point this was considered to be fairly unlikely, as that would make opening up a competing world extremely easy. (They did announce it officially soon after, but it has yet to actually happen, and probably won't anytime soon). Rather than wait years for an official release, Darren Guard founded the OpenSim project, to create an open source virtual world server to which any Second Life client can connect**. An alpha release soon became available, and the project continues to improve to this day.
The software is by no means complete, and in a few ways, it still lags behind Second Life. Moving between sims* can be buggy, and from my experience, the whole system was generally less stable. However, it has numerous advantages that come simply from being open source and free.
The biggest advantage is that anyone can own a server. Most modern computers are more than fast enough to run as an OpenSim server, so anyone can set up their own grid. Several groups already have, providing a number of alternatives to the main Second Life grid*. You could set up a grid, probably in less than an hour, provided you were willing to read the guides, and consult Google if necessary. This makes setting up your own projects much cheaper and easier than in Second Life.
It also allows for much more customization. OpenSim can run on nearly any hardware. You can decide whether to allow anyone, to block everyone except yourself, or anything in between. Communities can decide what features they want to allow, and what ones they want to block. This makes the system ideal for a number of group and individual uses, where the trolls and chaos of Second Life would prove inconvenient.
The most obvious uses are social, corporate, educational, and generally based around large groups, but there are a lot of practical uses that an individual can find for this software. Are you an artist? There are leagues of untapped artistic ground in designing your own worlds, and this is probably the cheapest way to do it. Are you about to move? You can throw together a very basic model of your new home, and use it to figure out where you can fit your furniture. Do you have a presentation to complete? Hosting it in a virtual world can make it much more impressive. In a high school psychology class, I was once required to present a creative model of the brain. Modeling it in OpenSim was very quick, and resulted in a memorable presentation. Generally speaking, creating a virtual world is very impressive to those not familiar with the software, and you can use this to your advantage in a number of ways.
Like Second Life itself, OpenSim is not the future of virtual worlds so much as a symbol of it. It's hard to imagine it becoming something that more than a few enthusiasts will use, but you can see how it's only a few steps away from being the kind of crazy scifi technology that we all want virtual reality to become.
*A sim is an area of virtual land 512 m^2. Several sims connected together are a grid. A grid can contain sims hosted on many separate servers, which can cause problems, if all the connections involved don't work as planned.
**Thanks to raincomplex for suggesting a better way of phrasing that sentance.