In the U.S.A., the Summer solstice is the first day of Summer. Elsewhere, the solstice is actually regarded by some as being the middle of Summer (or Winter). A Midsummer Night's Dream is supposed to occur on the Summer solstice - and that's midSummer.
The moment when Earth is closest to the sun is called perihelion. It occurs a few days after Winter solstice (in the Northern hemisphere). When the Earth is farthest from the sun is aphelion, and it's a few days after (Northern hemisphere) Summer solstice. The solstices are not the days when the Earth is nearest or farthest from the sun, they are the days where the poles most closely point towards the sun (or, when the poles are farthest from the terminator; or, when the celestial equator is farthest from the sun). To think they are the same event is a fairly common misconception, because the solstices are typically the warmest and coldest days of the year. However, the angle of the land to the sun, and the number of hours the sun shines, have a much larger effect on the temperature than the relatively small variation in distance caused by the slight eccentricity of Earth's orbit.
As is noted in the Summer solstice node, the sun is at zenith at some point on the Tropic of Cancer. All points between the Equator and the Arctic circle experience the longest day. All points between the Equator and the Antarctic circle experience the shortest day. (Within the Arctic and Antarctic circles, days get quite unusual, so I'll leave that for another writeup.)
As is noted in the Winter solstice node, the sun is at zenith at some point on the Tropic of Capricorn. All points between the Equator and the Arctic circle experience the shortest day, and all points between the Equator and the Antarctic circle experience the longest day.
Daytime and nighttime on the Equator are both 12 hours long, all year.
It is interesting to note that, for points between the Tropics and the poles, the Summer solstice is the same day the sun most closely approaches zenith -- however, for latitudes between the Tropics, the sun reaches zenith twice in a year, and these moments do not correspond with the solstices. In fact, the sun hits zenith at the Equator on the equinoxes, about halfway between the solstices.