A mobile phone is a telephone that can be carried from place to place, generally small enough to fit in one's pocket (though some early mobiles were much larger than that), and does not need to be connected to any fixed infrastructure, using radio waves to transmit the call.
Such phones have existed at least since the 1950's, though the distinction becomes blurry when special systems are used to "patch" conventional two-way radios into a phone network with the assistance of human operators. Modern mobile phones can make and receive calls totally automatically, operating as would a normal phone (though most have a superset of the ability of fixed-line phones).
Mobile phones began to proliferate through the 1980's with the introduction of "cellular" phones, with multiple base stations located relatively close to each other, and protocols for the automated "hand-off" between two cells when a phone moved from one cell to the other. In this era, mobile phones were somewhat larger than current ones, and many were designed for permanent installation in cars, or as "transportable" phones the size of a briefcase.
As technology improved throughout the 1990's, the larger "bricks" disappeared and tiny hand-held phones became the norm.
There are two main technologies used in mobile phones: cellular technology, and satellite technology.
Early mobiles were analog; newer ones are digital.
Mobile phone technology is often divided into generations: 1G, 2G, 2.5G, 3G:
All of these technologies were based on cellular technolgy. However, satellite based phones are called mobile phones too.
Many mobile phones support 'auto-roaming', which permits the same phone to be used in multiple countries. However, both countries must use the same mobile system, the same frequencies and there must be an agreement between the two countries' telephone operators.
Mobile phones must be distinguished from portable phones (called cordless phones in the US); with a portable phone the user purchases their own base station, which they connect to a landline, and the range of the phone is generally restricted to one building or so.