Image: diagram showing the shift of the Earth's axis with respect to the sun as it rotates on its orbit. Copyright and source: NOAA / NASA SciJinks via The Old Farmer's Almanac.
The term Equinox comes from the Latin “aequus” (“equal”) and “nox“ (“night”), as it means that day and night have same duration, 12 hours.
The equinox occurs twice a year, in Spring and Autumn, when the Sun rises and sets at the exact east and west. To define exactly the length of the day, the dawn begins when the sun has passed the mid-horizon and the sunset ends when the Sun is half below the horizon.
In the northern hemisphere, the equinox of March is called the Vernal (Spring) Equinox and September Equinox is called the Autumn Equinox; in the southern hemisphere is the opposite. As the axis of rotation of the Earth is tilted with respect to the orbital plane, even the celestial equator is inclined to the Ecliptic, creating 2 intersections, just the equinoxes, where the Sun crosses the Earth's orbital plane or equatorial plane. So, the equinoxes are not days but they are points, well-defined moments.
The Spring Equinox is also called First point of Aries, the Autumn Equinox is also called the first point of Libra (ω). Due to the precession of the equinoxes (see section later) these equinoxes are no longer located in the constellation from which they are named, but they have shifted to another constellation.
At the equinoxes, in the places of the Earth between the lines of the Tropic of Cancer and Capricorn, i.e. the Tropical belt, the Sun is directly overhead, perpendicular to the observer.
Note: Easter falls on the first Sunday after the first full moon after the March equinox, where for church Spring equinox is falling always on 21th March. The earliest possible Easter date in any year is therefore March 22 on each calendar. The latest possible Easter date in any year is April 25.
The term Solstice comes from the Latin “sol” (“sun”) and “stitium” (“stoppage”, “stand still”), so, the sun-standing.
Every day at midday the sun rises to the maximum height above the horizon. This height is changing day by day with slight variations, and again is at maximum at the Summer Solstice (June). Before the Sun had risen in the sky, thereafter the Sun moves back down again until the minimum height in the year for the Winter Solstice (December).
As consequence at Summer Solstice the days are at longest, while nights at shortest. The opposite is valid for the austral hemisphere.
The Sun is at its zenith in June in the places crossed by the Tropic of Cancer, in December in the places crossed by Tropic of Capricorn, while outside of the tropics it reaches the maximum possible height above the horizon to that latitude.
In Astronomy the Solstice is defined as the day when the Earth is located in one of the two most distant points of the Ecliptic from the celestial equator, or rather when the Sun passes through the northernmost point of its path along the Ecliptic. In the same way the Winter Solstice is the moment when the Sun passes through the southernmost point of its path along the Ecliptic.
At Winter Solstice the low position of the Sun in the sky, together with the low inclination of the rays causes the low temperatures typical of winter. In the southern hemisphere at the same time the sun lights and warms to a greater extent, as the days are longer, and its rays are nearly perpendicular to the surface.
Please note that the changing of the seasons is due to the inclination of the Earth and not to distance from the Sun, as in the northern hemisphere the hot season coincides with the period of maximum distance from the Sun.
Solstice similarly as Equinox indicates not just the exact point, but the whole day, the longest in the year.
Here below a diagram illustrating the concept of seasons on Earth (copyright and source: NASA Space Place via Wikimedia Commons), and an animation of the Earth as seen daily from the Sun looking at UTC+02:00, showing the solstice and changing seasons (copyright and source Joost99 User / Wikimedia Commons):