I've seen a lot of confusion about what a solstice is today. I've spent, well, honestly more time than I should have trying to clear up the confusion. Honest mistakes are one thing, we all make them now and then, but some people would rather eat their own faces than admit they might be wrong, apparently...
So yeah, let's start this talking about what a sabbat is. The term sabbat comes from Gerald Gardner and Wicca, although it's spread to many neo-pagan traditions these days. There are, to most who follow this system, eight sabbats in a year. Two solstices, two equinoxes, and four cross quarter days. (Esbats are another beast altogether, and these days usually refer to a full moon celebration, or other lunar celebrations.)
I think for some the confusion is simply mixing up the word sabbat and solstice. Truly though, there are only two solstices in the year - one in June, and one in December. In the northern hemisphere these days mark the astronomical beginning of summer and winter, and fall around the longest and shortest days. (In the southern hemisphere, this is reversed, with the June solstice being the winter solstice, and the December solstice summer.)
If you want to read a bit more about the two solstices, take a look at this webpage. You can see in the last section the mention of equinoxes. Again, there are two of these in the year, one in March, and one in September. The word equinox is derived from a word meaning equal-night. In other words, on these two days the length of day and night are both equal to each other, or close to it. (The actual day this happens in your area might not be on the day the astronomical equinox occurs, but it will be very close.
Now, these four days, the two equinoxes, and two solstices, are sometimes called the quarter days. If you draw a circle, then draw a line from north to south, and one from east to west, you'll see why. Each point where a line touches the circle can represent one of these days, and they seem to cut the year into fourths, or, in other words, they quarter the year.
So, if you then draw an X through the + in your circle, you can see how there are four more days that cross the quarters, or, in other words, the cross quarter days. These are the sabbats Imbolc, Beltane, Lammas, and Samhain. (Or by any other variant name/spelling one might use for these days.)
And here's what is probably the worst Wheel of the Year graphic out there, but hey, hopefully it gets the point across.
Okay, now why does it matter? What's the problem if people want to call everything a solstice? Well, I mean, would you call an apple a cheeseburger? Would you call a bucket of ice cream a cheeseburger? What about a bowl of cereal? No, probably not, because that wouldn't make much sense. And if you did, people would probably get confused real fast. It is essentially like calling Halloween, Easter, and all other holidays Christmas. You might know what you're talking about, but other people are going to have no clue. So if you're, you know, online for the sole purpose of talking to people, that making sense bit is rather important.