Molecular Beam Epitaxy (also known as "MBE") is actually a very simple concept with a very complicated implementation.

At its core, one has a substrate of some material on which one wants to grow another material, generally a nanostructure. This substrate is placed in the MBE chamber, which is in ultra-high vacuum (involving cryopumps). The MBE chamber has attached to it a variety of side containers closed off with shutters. These containers hold elements such as Al, Ga, As, Se, Ge, and so on. The elements are heated, causing them to sublimate and emit atoms of that element. If one wishes to grow something containing a specific element, the appropriate shutter is opened for the desired time period, and thus emitting a beam of atoms onto the substrate. Using this method, one can get extremely fine--down to the monolayer level!--control of nanostructure growth.

There are more complexities, involving the solubility of different materials and annealing, but that's the subject of other nodes.

(Very) simplified illustration of an MBE machine (except that a real MBE is much, much, more round):
   +----+        |     ......++      |
   |    +--------+  ...      ||      |    Al: Aluminum source
   | Al          \           ||      |    As: Arsenic source
   +-------------+.....      ||<--SS |    SS: Substrate
                 |    ....   ||      |    / : shutter
   +-------------+....       ||      |    \ : shutter
   | As          /           ||      |    ..: edges of atom spray
   |    +--------+......     ||      |
   +----+        |      .....++      |