Basically, the Universal Time systems are coordinated world-wide standards, whereas GMT was a national standard that was used as a de facto world standard.
Greenwich Mean Time is the mean solar time plus 12 hours as observed from Greenwich, England. (Mean solar means that you use average out the seasonal changes due to the earth's elliptical orbit by using a theoretical mean sun rather strictly observing the real one. Otherwise day length would vary by over 1% throughout the year.)
But by 1972, atomic clocks had spread around the world and were used as centralized time-keeping fixtures. Around this time, a second was also redefined in terms of electron transitions in a cesium atom, rather than a certain fraction of a mean solar day.
This lead to the establishment of a coordinated atomic clock time called Tempes Atomique International (TAI).
But the earth doesn't always spin at the same rate... this is due to a combination of factors including polar eccentricity, periodic seasonal variations due to the ellipical orbit around the sun, the slowing down of the earth (that is, momentum transfer to the earth-moon system), plus other unexplained empirically observed changes.
So this "pure" theoretical time (TAI) would drift farther and farther out of alignment with the real world.
So a new standard called Coordinated Universal Time (UTC) was created. It is TAI with an offset of a certain number of leap seconds, currently 22 (though there was a 10 second difference between TAI and UTC due to a variety of slippages). These leap seconds adjust the atomic-clock time back into the actual observed mean solar time with respect to the prime meridian.
For what it's worth, UTC is in fact just one of several of the Universal Time family of time standards. And if you want to be more accurate than tenths of seconds, you should also specifiy which UTi you are using... UT0, UT1, UT2, and so on. And since UTC is actually determined from each individual atomic clock, to be perfectly pedantic you should also specify which atomic clock you are using, such as UTC(USNO).
Of course, in practice, only a very, very few people have any real reason to be that anal. The rest of us can muddle along saying "quarter to four." :-)
Some more in-depth explanation is available from the NIST Time & Frequency pages.