Today is the winter solstice — specifically, it occurs at 23:38 GMT, or 6:38 p.m. Eastern (US) time. Technically what this means is that, at that moment, the center of the Sun is at the lowest declination of the year.
Astronomers measure the positions of objects on the sky using a coordinate system based on longitude and latitude on Earth, but to make it more confusing we call it Right Ascension and declination. The first measures the east/west position, and the second north/south. So just like an island might have coordinates of so-and-so degrees longitude and this-and-that degrees latitude, a star has an RA and a dec (if you want to sound cool and use astronomy slang).
OK. But what if an object moves? Well then, the coordinates change with time, too (just like the lat and long of a ship steaming across the ocean has changing coordinates). As the Earth orbits the Sun, the Sun’s position against the background stars changes, and therefore the coordinates change. I won’t go into details — you can read about it here, for example — but the Sun makes a sine wave across the sky over the course of the year, sometimes farther north, sometimes south. The time when the Sun’s position (actually, the center of the Sun’s disk) is at its farthest south — -23° declination — is called the winter solstice.
And that’s where we are today. The declination of the Sun has been getting lower ever since last June, and today it reaches its farthest point south. At 11:38 p.m. GMT, the Sun’s movement south stops (solstice literally means "Sun stops/stands still"), and it starts to slowly creep back north again. That’s why there is an actual moment, a point in time, for the solstice. In June it’ll get as far north as it can, and the process reverses. Incidentally, the times halfway between the solstices when the Sun’s declination is exactly 0 are called the equinoctes (the singular is equinox). Just so’s you know.
I’ll note the eclipse last night had nothing to do with the solstice at all, despite the hype. It’s just another day of the year as far as the Moon is concerned, so we get eclipses on the solstices every now and again, just like any other specific day. Heck, for me in Boulder, the first half hour of the eclipse was on the day before the solstice, and people on the west coast had the eclipse going for an hour and a half before midnight changed the date! That just shows you that we sometimes get excited about arbitrarily forcing our units (days starting at midnight) on the Universe, when honestly it couldn’t care less.
That doesn’t make the solstice any less cool, though. Every day now for people in the northern hemisphere the Sun will stay up a little bit longer, and days will stretch out, and things will warm up once again. It’s a time of renewal, which is no doubt why so many ancient cultures celebrated it.
And for you folks in the southern hemisphere? Well, take the inverse of what I said: the Sun is at its highest point in your sky now, the days will get shorter, and it’ll get colder. But never fear! Things will turn around once again, as they always have, and they always will.