Adam,
There is an easy necessary condition for using equilateral triangles: The angle between the sides of the given regular polygon must be a multiple of 6o degrees (so that the triangles can fill a corner). That leaves the regular hexagon as the only example.
For the more general question, one can always connect the center of the given regular ngon to the vertices, and that divides the ngon into n congruent isosceles triangles. I assume that you are thinking of a more interesting example. That's a topic that is very difficult, but has attracted a great many amateur mathematicians. There is a new book that tells the story: Greg N. Frederickson, DISSECTIONS: PLANE AND FANCY (2003). The author keeps a web page,
http://www.cs.purdue.edu/homes/gnf/book.html
It might be easier for you to find an older yet important book on the subject: Harry Lindgren, Recreational Problems in Geometric Dissections and How to Solve Them. It was published originally in 1964. (There was a Dover reprint around 1972).
Chris
