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 n-gon to the vertices, and that divides the n-gon 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,
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).