Epoch simply means date and the term is used for specifying the reference date and time on which to base an object's coordinates within the celestial sphere, e.g., a star's. The current standard epoch is the J2000 epoch (aka J2000.0 epoch or epoch J2000.0), January 1, 2000, 11:58:55.816 UTC. Thus, if you specify that epoch, you mean the object was at the given coordinates at that time.
Previous epochs used in astronomy have been B1950.0 and B1900.0 with years 1950 and 1900. J stands for Julian date and B stands for Besselian date, which are based on a different specifications of the length of a year.
The difference between epoch and equinox:
Epoch gives a time when something was at a particular coordinate. Stars are moving relative to the Earth and Sun, showing proper motion, i.e., always changing coordinates, and for sufficiently-near stars, this change is easily noticeable as position-records are kept.
Equinox is a time used in specifying the coordinates themselves. Coordinates are based upon the position of visible astronomical bodies (e.g., the Sun), but these move relative to us with time in complicated ways, e.g., cycles. The equinox time specifies at what date/time we're using the positions to specify the coordinates. In particular, equatorial coordinate systems use the intersection of two planes, that through the celestial equator and that through the ecliptic. These change over time, e.g., the celestial equator follows the Earth's axial precession, so to fully specify a celestial coordinate system, you specify the time.