GMST - Greenwich Mean Sidereal Time
Sidereal time is the measure of the earth's rotation with respect to distant celestial objects. Compare this to UT1, which is the rotation of the earth with respect to the mean position of the sun. One sidereal second is approximately 365.25/366.25 of a UT1 second. In other words, there is one more day in a sidereal year than in a solar year.
By convention, the reference points for Greenwich Sidereal Time are the Greenwich Meridian and the vernal equinox (the intersection of the planes of the earth's equator and the earth's orbit, the ecliptic). The Greenwich sidereal day begins when the vernal equinox is on the Greenwich Meridian. Greenwich Mean Sidereal Time (GMST) is the hour angle of the average position of the vernal equinox, neglecting short term motions of the equinox due to nutation.
In conformance with IAU conventions for the motion of the earth's equator and equinox [ref 7] GMST is linked directly to UT1 through the equation
GMST (in seconds at UT1=0) = 24110.54841 + 8640184.812866 * T
+ 0.093104 * T^2 - 0.0000062 * T^3
where T is in Julian centuries from 2000 Jan. 1 12h UT1
T = d / 36525
d = JD - 2451545.0
It might seem strange that UT1, a solar time, is determined by measuring the earth's rotation with respect to distant celestial objects, and GMST, a sidereal time, is derived from it. This oddity is mainly due our choice of solar time in defining the atomic time second. Hence, small variations of the earth's rotation are more easily published as (UT1 - Atomic Time) differences. In practice, of course, some form of sidereal time is involved in measuring UT1.
LMST - Local Mean Sidereal Time
Local Mean Sidereal time is GMST plus the observer's longitude measured positive to the east of Greenwich. This is the time commonly displayed on an observatory's sidereal clock.
LMST = GMST + (observer's east longitude)