Short for accelerated processing unit or advanced processing unit, APU is any processing that happens outside of the computer's CPU. For example, the GPU (graphics processing unit) is an example of an APU, which can be used for more than rendering graphics.
By offloading tasks that would be only performed by the CPU, computers that take advantage of APU can have a significant performance increase.
AMD's Piledriver core or Trinity chip is a good example of a chip designed as an APU. This chip integrates a CPU and a GPU on the same die, which increases data transfer and lowers power consumption. An APU can also be such things as accelerators and FPGA.