An amplifier consists of two main sections, the preamp and the power amp. The preamp out allows you to take the output of the preamp and send it to line level effects, direct input to boards, or other power amps with no effect on the operation of the amp. The power amp in disconnects the preamp from the power amp, and connects whatever you plug into it to the input of the power amp.
So, say you needed more volume. You get a Sunn Concert Slave amp and another speaker cabinet. You run a cord from the preamp out to the Slave, and the slave to the second speaker. Your controls on the front of the Concert Lead control both amps.
Or, say you have a line level effect, like a delay line, and you want to delay the sound after all of your effects and tone controls. you run a cord from the preamp out to the effect, then from the effect output to the power amp in. Now, the signal the power amp amplifies will be the output of the effect unit.