Braising is one of those cooking terms that sounds intimidating, but is actually rather simple and easy to learn. The basic concept is to first saute your food and then allow it to simmer until it is completed, combining the benefits of dry and moist heat.
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Of course, the biggest benefit of braising is that it can make your food infinitely more delicious than sauteing or simmering alone could ever do. The reasoning for this flavor enhancement is simple science. By braising, you allow the food to brown first, creating the delicious, deep flavors caused by the Maillard reaction. By simmering in a liquid (whether added to the pan or created during the initial searing), the food then takes on the depth and complexity of whatever liquid is used. Additionally, this slow-cooking style allows the cells of meat or vegetables to break down gradually, creating a remarkably tender texture.
As for how to braise, well, this infographic from Illustrated Bites should help you master this technique in no time.