ARM = Advanced Risc Machines (previously Acorn Risc Machine). In 2013, 60% of mobile devices contained ARM-based chips. It now has a monopoly on handheld devices.
ARM is a family of RISC (Reduced Instruction Set Computer) architectures.
ARM Holdings Inc. is a fabless semiconductor company: they design chips and the license their technology to semiconductor producing companies such as ATMEL, NXP or Samsung.