Sidereal time is time reckoned by sidereal days and is the time elapsed since a transit of the vernal equinox, this is, strictly, apparent sidereal time. Mean sidereal time being reckoned, not from the actual transit of the vernal equinox, but rather from the transit of the mean vernal equinox (see: mean sun).
Generally, time is based off the apparent motion of the stars. Normal clocks follow the motion of the Sun: whenever the clock says 1 p.m. the Sun is high in the sky. The Sun seems to circle the Earth in about 24 hours. This is not true with sidereal time.
If, on a certain night, at 23:25 you see a star over a hill, next night it will be seen at the same place at 23:21. A sidereal clock is a clock which gains four minutes a day, so if a star is seen at 13:22 sidereal time, it will appear at 13:22, sidereal time, the next night.
To make a sidereal clock, imagine that in a year (a mean tropical year) the Sun turns 365.2422 times around the Earth, while the stars turn exactly once more, 366.2422 times. Compute from this the number of seconds the clock must advance each day, and make the clock.
Now place a string horizontally, following the meridian (north to south) and look at the stars from exactly below it. Wait for a known star to cross the string. Look for the right ascension of the star in any star map. If the map says that star is at 14 hour 45 minutes right ascension, set your sidereal clock at 14:45. From then on, when your sidereal clock says it's 16:00, then the line of sky you see behind the string is the same line marked in the map as "right ascension 16:00". So, to measure the position of a star, the Moon or the Sun, just wait for it to pass behind the string, and then take a look at the sidereal clock.
Up to a few decades ago such "sidereal clocks" were in fact built and used at observatories. Nowadays astronomers use other time scales to timestamp their observations, and "sidereal time" is now just a fancy name for "the right ascension of the point at the zenith".
Stars don't return exactly to the same place after one sidereal day because of the precession of equinoxes and other minutiae.