In simplest terms, an equinox represents a day with equal duration of day and night (Mar 21 and September 23) and thus we have a spring and a fall equinox. And solstice refers to a day with either the longest day (June 21, also called summer solstice) or shortest day (Dec 21, also referred to as winter solstice). On summer solstice, the north pole points directly towards the sun and on winter solstice, the north pole points directly away from the sun. In other words, on the day of summer solstice, sun is directly above the Tropic of Cancer. And on the day of winter solstice, the sun is directly above the Tropic of Capricorn. On the days of equinox, the axis of rotation of earth is perpendicular to the line joining earth and sun. Another characteristic of equinox is that on these days, the sun is directly overhead on the equator at noon.
The movement of earth around the sun is the reason for equinox and solstice, and this motion is the reason for the occurrence of seasons on earth.
Hope this helps.