Equinox is a date when the duration of the day is equal to the duration of the night. The Sun will pass directly above the equator, bringing nearly equal amounts of day and night to all parts of Earth. At the equator, an equinox results in about 12 hours of daylight and 12 hours of the night. Equinox means "equal night" in Latin.
Solstice, is an event that occurs twice a year, where the Sun appears to reach it's northernmost or farthest point from an observer on earth or southernmost or the closest point from an observer on earth.
An equinox occurs midway between the two solstices, due to the position of the sun, solstices are dates when the duration of the day is least equal to the duration of the night.
The Summer Solstice is the longest day of the year and happens around 21 December in the Southern Hemisphere and 21 June in the Northern Hemisphere. The Winter Solstice is the shortest day of the year and happens around 21 June in the Southern Hemisphere and 21 December in the Northern Hemisphere.
In simple words, if today is the equinox, every day from tomorrow will have fewer equal hours during the day in relation with the night until the next equinox.
Both equinox and solstices belong to "astronomical" seasons that occur due to the tilt of the Earth's axis. Like there are two solstices, Summer and Winter Solstices, there are two equinoxes, the Spring and Fall Equinoxes. One around March 21 and another around September 22, the March equinox is also called as the vernal equinox signaling the onset of spring and The September equinox is also called as the autumnal equinox signaling the onset of Fall.
The names of the equinoxes are reversed depending on which hemisphere you experience it!