The Maillard reaction (pronounced "my-YARD") is a culinary phenomenon that occurs when proteins in meat are heated to temperatures of 310°F or higher, causing them to turn brown.
Named for the French chemist Louis-Camille Maillard who discovered the process at the start of the 20th century, the Maillard reaction is similar to the process of caramelization, where carbohydrates like sugar turn brown when heated.
Note that while caramelization is not the identical chemical process as the Maillard reaction, only the most pedantic hairsplitter would actually take the trouble to correct you on it. Also simply referred to as "browning."
The Maillard reaction is the principle behind the browning of meat when it is seared as a prelude to braising it. This process creates a thick, dark-brown crust on the surface of the meat that enhances its appearance and flavor, and can only be created by high-temperature, dry-heat cooking techniques.
Note that one of the prerequisites for obtaining a dark brown crust on your meat is ensuring that it is dry before you put it in the pan. If it's too wet, the moisture will cook off, forming steam, which interferes with the browning process and tends to produce a gray exterior rather than brown.
Also, make sure that you get your pan very hot before adding the meat. A cast-iron skillet (like this one) is excellent for browning meat because it gets very hot and maintains its temperature very well.
See Also: Does Searing Meat "Seal In" Juices?