To me, oscars are THE prototypical "pet" fish. I think it's partly their personality, partly their size, and partly the fact that they're so often kept just a few in a large tank.
Koi are another that are very personable, often swimming right into your hand.
Big saltwater angels are easy to get attached to, too. Very charismatic (for a fish!), beautiful, lots of personality.
For planted tanks, though, angels, and dwarf cichlids. I've got rams now, but have had kribs, brichardi, julies, curviceps, and keyholes in my planted tanks over the years, too. I tend to keep medium-sized tanks, and a pair of dwarf cichlids is almost always my certerpiece.