If you were to measure the power dissipated by a chip over a time period, the dissipated power vary at each instant of time. Peak power refers to the maximum power drawn by the circuit. As this is a surge in power causing a surge in current drawn from the power source, this can cause glitches in the circuit. Hence reducing the peak power is essential for reliable functioning of the chip.
Average power refers to the average of the instantaneous powers dissipated over a time period, T. Calculating this number gives us the power dissipation of the circuit over the period of its operation, and helps determine the packaging, cooling and battery requirements. If your average power is high, your battery life is reduced.
Reducing the average power will automatically reduce the peak power.