An apparent sidereal day is the time it takes for the Earth to turn 360 degrees in its rotation; more precisely, is the time it takes the vernal equinox to make two successive upper meridian transit. This is slightly shorter than a solar day; there are 366.2422 sidereal days in a tropical year, but 365.2422 solar days, resulting in a sidereal day of 86,164.09 seconds (or: 23 hours, 56 minutes, 4.09 seconds).
The reason there is one more sidereal day than "normal" days in a year is that the Earth's orbit around the Sun cancels one sidereal day out, giving observers on Earth 365 (and a quarter) days, even though the planet itself rotated 366 (and a quarter) times.
Midnight, in sidereal time, is when the vernal equinox crosses the upper meridian.
A mean sidereal day is reckoned, not from the actual transit, but from the transit of the mean vernal equinox (see: mean sun).
See also: sidereal month, sidereal time, time