The term arises from the traditional use of circuitss at airfields - a landing aircraft will first join the circuit pattern and prepare for landing in an orderly fashion. If for some reason the pilot decides not to land, he can simply fly back up to circuit height, and complete another circuit - in other words, go around again. The term go-around is still used even for modern airliners, that do not use traditional circuit patterns for landing.
A pilot may elect to go around for many reasons, and in itself this does not constituite any sort of emergency. Passengers of commercial aircraft should not be alarmed if this happens to them when landing - often it is simply the result of an earlier aircraft not clearing the runway sufficiently quickly, or the pilot feels unhappy with a manual approach.
Many modern aircraft such as the Airbus series, which use fly-by-wire systems have automatic go-around modes. The pilot simply presses a button and the aircraft will take all necessary steps to abort the landing. In a more basic aircraft such as those found in general aviation, the pilot performs the go-around manually. This involves: