For the purposes of mathematically modeling the rules to the Game of Life (which would be useful if, for example, you want to program your own version of the game), I humbly present the following:

Given that the game is played out on a two-dimensional grid, let f(x,y) be the function describing the state of the grid point (x,y).
If f(x,y) = 1, then the point is alive. If f(x,y) = 0, then the point is dead.

Let N(x,y) be the number of neighbors of (x,y) that are alive. In other words:
N(x,y) = f(x-1,y-1) + f(x-1,y) + f(x-1, y+1) + f(x, y+1) + f(x+1, y+1) + f(x+1, y) + f(x+1, y-1) + f(x, y-1)

The function f'(x,y), which describes whether (x,y) will be dead or alive on the next turn, can be defined as:

          {  0       if N(x,y) < 2
f'(x,y) = {  f(x,y)  if N(x,y) = 2
          {  1       if N(x,y) = 3
          {  0       if N(x,y) > 3