How many ways are there to put 4 distinguishable balls into 2 distinguishable boxes?
For each ball, there are 2 choices of which box to place it in.  Since this choice is independent for each of the 4 balls, we multiply the number of choices together.  Hence there are $2^4 = \boxed{16}$ ways to place 4 distinguishable balls into 2 distinguishable boxes.