Phone answering machine, answering machine or just plain answer phone! Whatever you choose to call them they are undoubtedly one of the most useful telecoms inventions of the 20th century.
Old fashioned phone answer machines used a tape to record both an outgoing message and store incoming messages on. One of the best enhancements in the last ten years or so has been the introduction of solid state memory which means never running out of tape, and more reliable, smaller and cheaper units.
Phone answering machines can either be standalone which connect to an existing phone, or, more commonly now, built into the base unit of cordless phones.