Let G be a (countable) discrete group acting by diffeomorphisms on a smooth manifold M. Assume that the action is smooth, proper, and co-compact. Let D be a G-invariant elliptic differential (or perhaps pseudo-differential) operator on M. What should we mean by the equivariant index of D? This talk will take up this issue and will explain how equivariant index is used in the BC (Baum-Connes) conjecture. An example is G = Z, M = R, the real line, and D = -id/dx. Z acts on R by the usual translation action. This example will be considered in detail. This talk is intended for non-specialists. The basic definitions will carefully stated.