From the Latinsol 'sun' + stitium 'standing still' the early term solstitium became shortened to solstice in the 14th century.
The term refers to the two points in the zodiac (0° Cancer; 0° Capricorn) where the Sun's latitude is at its greatest distance north or south of the equator. The Sun's movement to these extreme points marks the peak of the seasons (mid-summer or mid-winter), so the Sun appears to stop rising or falling in latitude, and temporarily halts (stands still) as it reverses its relationship with the equator.
The Sun's placement at 0° Cancer marks the northern hemisphere summer solstice, which occurs around 21 June; it's placement at 0° Capricorn marks the northern hemisphere winter solstice, which occurs around 21 December (the seasons are reversed in the southern hemisphere).
Contrast with the equinoxes: the two moments in the year when the sun's path falls upon the equator.