Greek yogurt is regular yogurt that has been strained once, or several times, so that some of the water drips out, leaving a thicker product behind. It can range from a little thicker than regular to about the thickness of sour cream.
You can make it yourself from regular yogurt with a coffee filter; if you strain it more, for example by leaving a cheesecloth bag of yogurt hanging for up to 12 hours, you’ll have a really interesting cheese product.
Is it healthier? Not really, as it’s the same yogurt. But it’s refreshingly thick, even if it’s made from skim milk, which feels good in your mouth and really takes to fruit well. From the store my favorite is Chobani, like the pomegranate flavor I had with my breakfast this morning.