A quad core processor is a processor that has four processors
The processor is the brain of your computing device. Until in recent years there was a single processor in personal computing devices. The processor does the thinking and calculations for your computer. When the processor runs it does one calculation/direction at a time between the different programs.
Dual core processors first appeared. In the dual core processor there are actually two processors built in the processor on the computer. With the dual core processor the machine can actually do two things at one time with one processor doing the steps at exactly same time as other processor works on another program. This speeds up the computer.
Think of a multi-processor as if you had two separate brains in your head, you could do thinking faster and on more thins at one time.
The quad core processor is quite common now. Speeds of processors have not increased a lot in recent years, but with the multi-processors that same speed (say 3 GHz) is much faster than the single processor of same speed. The quad core will be effectively almost 4 times as fast as the single processor.
There are eight core processors available now but still on leading edge of technology. Larger machines may have even larger numbers of processors.