The most common shape memory alloy is called nitinol.  Equal amounts of nickel and titanium are combined and melted in an electric furnace at 1300 degrees.  The process is carried out in a vacuum to avoid contamination. Ingots are poured, and then either rolled into sheets or very fine wire.
A sequence of heat treatment is used to “program” the material to a specific shape.  For example, a length of wire can be made to remember that it should straighten out if heated above 70 degrees.  If you bent this wire …