He did run for a second term, but he lost to Jefferson (by a very small margin, though). Adams' unpopularity only existed at the start of his presidency. When the American people realized how good his decision was to keep America out of war with France, though, his popularity increased dramatically (but still not quite to Jefferson's level).

He was a highly intelligent man, having been a lawyer and writer, having helped draft the Declaration of Independence with Jefferson, having single-handedly arranged massive loans from the Netherlands to America during the war, and having been America's diplomat to France and then Britain. He was one of the most distinguished and hard-working Fouding Fathers, as well as one of the most principled (he and his wife both strongly opposed slavery and never owned a slave in their life - with his wife Abigail even freeing one that was given to her).

The unpopularity early in his presidency was due to his personality (many saw him, somewhat accurately, as being pompous and a bit arrogant) as well as him endorsing the Alien and Sedition Act, which was highly controversial among the pro-French citizens of America. But ultimately it was the realization that he'd spared America from a bloody war with France against Napoleon that "redeemed" him to many Americans, but this came too late for it to give him a second term, and Jefferson was immensely popular, in no small part because he had a charisma that Adams just didn't have.