Sous-vide is a french term for "under vacuum" and it essentially involves cooking food in vacuum seeled bags in a water bath. Cooking times vary, but usually you keep the water well below boiling and instead cook it around 130-140 degrees for a long time. Sometimes the cooking can last for a few days before you unseal it.
The process was originally designed in 1799, but it didn't really gain modern popularity until recently. The reason is pretty simple. The tools to make a great sous-vide dish weren't available until recently. now-days, with a vacuum sealer and a high-end thermal immersion circulator. Of course, if you're like me and don't feel like having all that expensive equipment around, an airtight ziploc bag and a rigged slow cooker will do. This low barrier of entry really helped increase the popularity, because now anyone can do it.
Why go through that effort? Pretty simply, it allows less flavor to escape through cooking. All the fat, juices, and spices remain in with the meat and really help penetrate. Further, it allows for much more even cooking. You won't have to worry about whether or not the heat went all the way through. It's also good for preparing a few meals at the same time for the future. In fact, sous-vide was used to produce a large quantity of meals for hurricane evacuees.
What sous-vide doesn't allow, however, is good browning or caramelization. The temperature simply doesn't get high enough. Perhaps the best way is to cook in the sous-vide then throw the meat onto the skillet on very high heat briefly just to add a good crust, caramelization and browning.
How long should you leave the food in the sous-vide cooker for? That really depends. Some food can become mushy if left too long, but left too short it can be unsafe to eat. Make sure to read up and make sure before you prepare your dish for best results.
Otherwise, feel free to try it out and enjoy!