Oh good, you know what a transistor is!
Basically your CPU is a processor designed to process general tasks and calculations. There are libraries of "instructions" which is just a list of the things you can ask it to do. You'll never use these unless you write assembly code for the CPU.
So, a CPU is designed to run programs. The cpu contains something called registers, which allow it to store small amounts of information. There are data registers and instruction registers. The instruction registers are very important because they tell the CPU what instuction it should follow. Next, there is a register called a program counter which contains information on where to find the next instruction. So, when the CPU computes the current instruction, it then loads new information from the program counter.
I mentioned data registers. The data registers are mostly changed by instructions, which either tell the cpu to get new data from the computers memory, or add data registers together to get new data.
That's the basic core logic, really simplified of course. If you want more info go to wikipedia. Also look up pipelining, and CPU cache.