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A microcontroller is by definition a is a computer on a chip. It includes all the necessary parts (including the memory) all in one IC. You just need to apply the power (and possibly clock signal) to that device and it starts executing the program programmed to it. A microcontroller generally has the main CPU core, ROM/EPROM/E2PROM/FLASH, RAM and some accessory functions (like timers and I/O controllers) all integrated into one chip. The original idea behind the microcontroller was to limit the capabilities of the CPU itself, allowing a complete computer (memory, I/O, interrupts, etc) to fit on the available silicon real estate. Microcontrollers are typically used where processing power isn't so important. More important are generally compact construction, small size, low power consumption and that those chips are cheap. For example controlling a microwave oven is easily accomplished with the smallest of microcontrollers. There are countless number of small electronic devices which are nowadays based on microcontroller. A modern home can include easily tens or hundreds of microcontrollers, as almost every modern device which has electronics has a microcontroller (or more than one) inside.

PIC microcontrollers from Microchip are very popular microcontrollers. PICs are easily programmable cheap microcontrollers. PIC16F84 seems to be the "standard" for small gadgets you see in the web projects.

The Basic Stamp II
Stamp In Class: collection of curriculum, hardware, and educational ideas about using the BASIC Stamp to teach electronics, microcontroller interfacing, robotics, physics, and interactive art.

The 8051

The 8051 is an 8 bit microcontroller originally developed by Intel in 1980. It is the world's most popular microcontroller core, made by many independent manufacturers. A typical 8051 contains CPU with Boolean processor, 5 or 6 interrupts, 2 or 3 16-bit timer/counters, programmable full-duplex serial port, 32 I/O lines, RAM and ROM/EPROM in some models. The 8051 architecture is quite strange and original. One strong point of the 8051 is the way it handles interrupts. Vectoring to fixed 8-byte areas is convenient and efficient. The 8051 instruction set is optimized for the one-bit operations so often desired in real-world, real-time control applications. The 8051 has the widest range of variants of any embedded controller on the market.

My 8051 Slides from Class

AT89C51 Datasheet

AT89S8252 Datasheet

The 8051 Programmer's Guide and Instruction Set

The 8051 Tutorial

Easy Programmer