Well, the actual term 'micro-' when speaking about computers is not often used nowadays. Obviously, a microcomputer is a small computer (compared to big supercomputers) that has a microprocessor (again, the term 'micro-' in this word is now eliminated and it is just called processor), central processing unit (CPU).
What is Difference between a Microprocessor and Microcontroller? The term microprocessor and microcontroller have always been confused with each other.
Your PC is an example of a microcomputer. Microcontrollers are computer chips that (if you look at its name it becomes obvious) control something.
Microcontrollers usually have a processing unit (which usually runs on a low clock speed), internal memory and input/output elements. Pretty much every electronic device has a microcontroller - starting from a game console or electronic clock and ending with personal computers. Microprocessor is also called the Central Processing Unit (CPU). A microprocessor is currently working in your PC and it's either from AMD or Intel (or any other processor manufacturer). Basically, a microprocessor is one of the essential elements of a computer, as it handles all the operations a computer performs. A microcontroller is a single chip, containing a microprocessor that has some non-volatile storage, an enlarged cache or dedicated RAM, and (usually) some common peripheral components (like a UART, pwm, or ADC) all on the same chip.
Litterally a 'computer on a chip' I would also point out that virtually all modern microprocessors have a microcontroller equivalent (you can, for example, buy P4 and PPC microcontrollers), with similar performance specification to their stand alone processor counterparts. However, due to both costs and the small processing demands of applications microcontrollers are used in (running a toatser, car engine, stereo, EKG machine, et cetera, et cetera, et cetera), smaller, lower powered micro controllers tend to be most common. As a point of clarification, while PCs do have micro controllers in them, their participation tends to be more along the lines of operating the optical drive's motors, while the CPU(s) and GPU handle the actual processing.
Do read the question (and answers) stevenh linked to, but your question is somewhat different. A Central Procession Unit (CPU) is the part of a computer that sequences and executes instructions.
Other parts in the traditional computer architecture are the memory and the I/O. In the stone age days of computers a mainframe's CPU's occupied multiple cabinets. Later a minicomputer's CPU occupied one or a few PCBs. The next step was to integrate a CPU on a single chip. That is what we call a microprocessor (uP). From there the development forks:.
the CPU-on-a-chip is made more powerfull (faster, parallel execution, fast execution of complex instructions like divide and transcendentals), a cache is added, more CPU's are combined in one chip, etc. This results in the (mainly Intel) super-microprocessors of today. a moderately powerfull CPU (more powerfull than those in a uC, but less than those in a desktop CPU) is combined with a small boot ROM and a set of complex peripherals, like a video/lcd subsystem, mpeg decoder, wired or wireless ethernet interface, USB intefaces, etc. To cerate a 'system-on-a-chip'.
These chips form the hart of modern set-top boxes and the small Linux systems (Beaglebone, Raspberry Pi, etc.) derived from them. The CPU is combined with memory and I/O on the same chip, creating a complete computer on a single chip.
This is called a microcontroller (uC).