A microprocessor is a CPU which is implemented on a single integrated circuit. A few special-purpose microprocessors have also included main memory and/or ROM in the IC chip, but generally (even now) this is usually not included - although for performance reasons, there are generally main memory caches on modern microprocessor chips.
Microprocessors have completely changed the ways in which computers are used. When CPUs were physically large, and built out of many smaller chips (or even individual physical transistors, before that), they were expensive. This more or less mandated their use in ways such as time-sharing, which allowed their costs and overheads to be amortized. Once they became very cheap, constructs such as personal computers, and later smart-phones, etc, became economically feasible.
The first microprocessors were relatively limited, in architectural terms; IC's of that era were not very large (in terms of the number of transistors they contained), which severely limited the complexity of microprocessors built on them. Contemporary microprocessors have no such limits, and contain tens of millions of transistors, enabling them to use complex techniques such as pipelines, superscalar processing, etc.
The use of these advanced techniques, added to the reduction in size (thereby reducing speed-of-light signal delays), allows microprocessors to have greatly increased performance over all earlier computers, including mainframes. Microprocessors have therefore made essentially all other types of computers obsolete.
The first widely successful microprocessor was the Intel 4004.
Before IC microprocessors were common, the term "microprocessor" was in rare use as a portmanteau of "microcode processor". Examples of this type of usage include BCC 500, Maxc, Alto, NORD-10, IBM's PALM, and the processor in the DV11 Communications Multiplexer.