eiy = cos(y) + isin(y)
i is defined as sqrt(-1)
Euler's identity, given above, is a wonderful and mysterious result. The identity binds geometry with algebra and often simplifies the mathematics of physics and engineering (see phasor for an example). In some sense Euler's identity is more a definition than a result--one could define eiy in other ways. However, the chosen definition is clearly the most useful one.
Complex functions have very nice mathematical properties if they are analytic. In fact, complex analysis is based on analyticity over regions of the complex plane, allowing for isolated singularities. Given the solid structure inherent in analytic functions, it is desirable to define ez, where z = x+iy, in such a way that it is analytic over the entire complex plane. Taylor's Theorem tells us that if a function f(z) is analytic over the whole complex plane (entire), then it can be written as Σ1/n! f(n)(0)zn. Since Taylor's Theorem must hold for y = 0, the power series in z must be the same as the real-valued power series in x. In other words, if we want ez to be entire then it must be given by Σ1/n! zn.
Given this reasonable definition for ez we can obtain eiy by plugging iy into the power series. Miraculously, the result is the sum of the power series of cos(y) and isin(y)!
A special case of Euler's identity, when y = π, is a very famous equation since it is composed of five fundamental, seemingly unrelated mathematical constants. This equation is eiπ + 1 = 0.