|MadSci Network: Astronomy|
Good question, Jenny.
The short answer is that stars become red giants when they run out of hydrogen to burn and begin burning helium. The expansion to the red giant stage only occurs for stars lighter that about twice the mass of our Sun, otherwise they follow a different evoutionary path.
There's an excellent website with good detail on the process at:
http://www.physics.gmu.edu/classinfo/astr103/CourseNotes/ECT ext/ch1 8_txt.htm#18.2.
Try the links in the MadSci Library for more information on Astronomy.