ARM, Acorn RISC Machine, was a 32-bit CPU designed by Sophie Wilson and Steve Furber of Acorn Computers of Cambridge, England. The acronym was later changed to mean "Advanced RISC Machines" after the original Acorn Computers company was no more. ARM is no longer considered an acronym however.
Acorn Computers used the 6502 8-bit CPU, e.g. in their BBC Micro, which started to get a bit aged around 1983. Wilson and Furber concluded after some testing that the Nat Semi 32016, the Motorola 68000, and other CPUs they tested were not very efficient in using the memory bandwidth that had become available at the time (they all performed at around 4Mbit/s). To avoid that CPU bottleneck Acorn wanted something better and started looking into CPU design. At first Wilson and Furber visited some traditional microprosessor plants and found large buildings with thousands of engineers. But, because Wilson was a huge fan of the 6502 processor (and has been called "the best 6502 programmer ever" by Personal Computer World's Guy Kewney), they visited Western Design Center where Bill Mensch (designer of the CMOS version of the 6502, the 65C02) was busy designing their next generation CPU. Seeing the very small scale of the operation Wilson and Furber thought "we could do that", went home, and designed what would be the 32-bit ARM CPU, now referred to as ARM1. It used 25000 transistors. The first samples were ready in April 1985.
ARM is most known for its low power consumption, and is the main reason the architecture became so popular with mobile devices where it has about 95% of the market. The designers were not trying to create a CPU with particularly low power consumption, that was not much of a concern back in 1983. When the first sample was plugged into a development board the CPU worked perfectly (quite an achievement for a first version), but Wilson noticed that the meter in series with the power line to the CPU showed zero, so something was wrong. It turned out that the development board had a fault, it did not provide power to the CPU itself, only to the logic chips around it. The CPU was running only on leakage from the logic circuits. In the end it turned out that the CPU only needed one tenth of a watt, but this was all accidental to the design even though it was what secured ARMs huge market today, with several billions of CPUs sold every year (in comparision Intel has been selling around 1 billion CPUs per year lately. ARM CPUs are manufactured and sold by a number of vendors, so the comparision to Intel's production is not entirely fair). The architecture now comes in a lot of incarnations, including 64-bit versions.
The ARM CPU was used as an extension CPU for the BBC Micro "Tube" interface. It was also used in the Acorn Archimedes personal computer, eventually running RISC OS. The licensing scheme came into use when Apple wanted to use the ARM design in its Newton, a precursor to Personal Digital Assistants, or PDAs. But Apple did not want to buy the ARM from Acorn, which was a competitor in the market at the time. Apple, Acorn, and VLSI Technology thus together founded a separate company, ARM Limited. The Newton did not sell particularly well, but then Texas Instruments licensed the ARM design for use in a GSM chipset for the mobile phone maker Nokia. This was the first design which combined an application processor and a baseband processor on the same chip. The design was very successful and Nokia had a huge segment of the mobile phone market at the time. The rest is history. (Note: Another source says it was Ericsson which, with the help of VLSI, integrated the ARM into a GSM chipset. As Ericsson was a main driver in getting GSM working this also sounds plausible).
When Jobs came back to Apple in the late nineties, one of the first things he did was to sell Apple's ARM Ltd. shares, as Apple was short of cash (and Jobs didn't even like the Newton). VLSI Technology does not exist anymore, and today ARM Limited and the ARM design is held by ARM Holdings. The company is still located in Cambridge, UK.
 According to Chris Shore of ARM Ltd, ARM isn't an acronym anymore: "ARM doesn't stand for anything. Officially, it stands for nothing. It used to stand for Advanced RISC Machines, which was the company's first name on its foundation. Prior to that it stood for something else – it stood for the Acorn RISC Machine, because the original architecture was actually designed by another company, called Acorn."