I love sequels! Some of my favorite all-time games are sequels, like Kingdom Hearts II and Shadow Hearts: Covenant. Sequels work when they take what you loved about the last game and make it even better, whether its the combat, or the story, or the characters, etc.
As for what constitutes a sequel, I think that it largely depends on the genre. For me, since the majority of the games that I play are console RPG's, a sequel to me is anything with storyline, or at least time-line carryover. For example, Shadow Hearts: Covenant is a sequel to Shadow Hearts because it starts (more or less) where the first game left off, and Shadow Hearts: From the New World is a sequel to that, even though it has none of the same playable characters, since it takes place in the same world at a different time. Conversely, Final Fantasy VIII is not a sequel to Final Fantasy VII, since it has new characters, a new world, a new story, and no time-based relationship (e.g. the FF VIII world is not the FF VII world 400 years in the future, it is a separate entity). A common name does not a sequel make.
As for other genres, perhaps my compatriots could provide their insight on the matter.