Greenwich Mean Time (GMT) is a term originally referring to mean solar time at the Royal Observatory, Greenwich where a system was first developed around 1850 for tracking time based on the rotation of the Earth. It is now often used to refer to Coordinated Universal Time (UTC) when this is viewed as a time zone.
Strictly speaking, UTC is not a time zone but an atomic time scale which only approximates GMT in the old sense. It is also used to refer to Universal Time (UT), which is an astronomical concept that directly replaced the original GMT.
In 1970 the Coordinated Universal Time system was devised by an international advisory group of technical experts within the International Telecommunication Union (ITU). UTC is the International Atomic Time (TAI, from the French Temps atomique international) with leap seconds added at irregular intervals to compensate for the Earth's slowing rotation. Leap seconds are used to allow UTC to closely track UT1, which is the mean solar time at the Royal Observatory, Greenwich.
The ITU felt it was best to designate a single abbreviation for use in all languages in order to minimize confusion. Since unanimous agreement could not be achieved on using either the English word order, CUT (coordinated universal time), or the French word order, TUC (temps universel coordonné), the acronym UTC was chosen as a compromise.
The difference between UTC and UT1 cannot exceed 0.9 s, so if high precision is not required, the general term Universal Time (without a suffix) may be used.
In casual use, Greenwich Mean Time (GMT) is the same as UTC and UT1. Owing to the ambiguity of whether UTC or UT1 is meant, and because timekeeping laws usually refer to UTC, GMT is avoided in careful writing.
edit Time in Greenwich, UK
In the community of Greenwich, GMT (in the form of UTC) is the official time only during winter (during summer the time in Greenwich is British Summer Time rather than "GMT").