A longyan (or longan if you're going by the Cantonese spelling instead of the Mandarin one) is small pitted fruit that grows in Asia. It has a hard brown skin with sweet white flesh and a pit. The longyan is about the size of a large grape, and is more sweet and less floral-tasting than the lychee. They're relatively easy to shell (easier than the rambutan and about on par with the lychee).
Translated from Chinese, "longyan" means "dragon eye." It is named this way because when the shell is taken off, the dark pit shows through the white flesh, resembling an eye. Popularity-wise, the longyan seems to be less-favored cousin of the lychee. Their flavors are very similar, but the longyan is smaller and not as pretty. Though I've seen plenty of lychee-flavored items (candy, soft drinks, etc.), I've never seen anything longyan-flavored that wasn't, well, a longyan.
Pick them up at your local Asian grocery store. An H-Mart or Great Wall would be a good jumping-off point. I recommend eating them slightly chilled during a hot summer day.