Well an easy place to start would be to name a few really good sequels and then maybe think about what I like about one or two of them. The following are films which for one reason or another I believe are some of the best sequels you can find, not all of them will have a 2 plastered on the end but they are definitely considered real proper official sequels. OK so here is my list Dawn of the dead, The Empire Strikes back, Aliens, Star Trek 2 The Wrath of Kahn. I am not saying that these are the only good sequels or that they are the best but they are all sequels I respect and feel don't disappoint fans of the original.
So taking Aliens as a starting point, what does it take from the original film? Well it takes the character of Riply and it takes the alien creature the Xenomorph, it takes everything you learned and know from the first film about both this woman and this type of monster and then applies them to a new situation. It is definitely a sequel and feels like a sequel because it is the continuing story/adventures of Riply. So what is different? Well the first film was very much a horror film, there was one Alien hence the films title Alien and it basically stalked the crew of a ship the ship was a commercial spacecraft so they were just regular working people with no military training, there were also no real weapons on board with which to defend themselves, where as the Alien with its claws, its large frame, predatory instincts and acid for blood was a born killer. In the sequel Riply becomes the civilian advisor to a team of Colonial Marines basically bad ass soldiers with all of the weapons you could dream of, hand guns, plasma rifles, assault rifles, grenades, body armour, helmet cameras, they are tooled up trained professionals, professionals who would have killed that one lone Alien with very little trouble. The problem is that in this case they are not set against a lone Alien, rather they end up facing a relentless hoard of Aliens, yes they shoot, kill and incinerate lots of Aliens but with everyone they kill more and more just keep coming. It would be hard to call this a horror film like the first one, I would be inclined to think of it more as a Science Fiction Action movie with Elements of horror in it, in fact it kind of feels like a War film at times sort of either a Vietnam movie or one of those old Western fort under siege flicks. Yet this is one of my favourite sequels and that is because it has a strong connection to the original film bringing back both the last survivor and using the same Monster but it also is not afraid to take these set pieces and do something new and interesting with them instead of being just a cash in rehash of what came before.
Star Trek 2 the Wrath of Kahn and The Empire Strikes back are even more obvious than this, they are set in the same world's and feature pretty much all of the main characters from there respective prequels. Yet if we look at Dawn of the Dead the sequel to Night of the living dead there is no central character from the first movie returning to feature in the sequel, all that is returning is the monster, the Zombies.and these guys look a lot different, a lot better really. The first film was in black and white yet this sequel was in colour. The same Director was at the helm of both films and they were both based on his vision which wasn't true in the case of Alien and Aliens, Aliens was more like someone new jumping in to the original guys toy box and playing with the toys in his own way. So you can argue that in one way Dawn is a truer sequel because its by the same Director yet Aliens has a returning lead character, which one matters more? Well that depends on your own personal view, I don't mind somebody new being at the helm, new people can bring new ideas and a fresh perspective but I also don't mind sequels that are set in the same world but don't directly follow a certain persons story, in some cases it can be the world itself which is the central character.
So I guess basically a sequel is a continuation, it can be a continuation of a world story or a persons story or a whole group of peoples stories, it can be any of these things and be a good sequel as long as its good. If its the continuation of a person or a groups story then they don't have to do the exact same thing in the exact same place, if its a continuation of a worlds story it doesn't have to even be set in the same part of that world as the last film. A sequel can even be a part of a slightly different genre, what it has to do though is feel somehow connected and deliver something which will interest those who were interested in the original. Often it is when a sequel is too similar or far too different that it is considered bad, people will moan if something is just a rehash but then they will also moan if it doesn't feel connected at all. I think the best formula would be to take elements of the past but to use them to do either a completely new story or a continuing story which offers your fans something of the old while not overly repeating yourself to the point people might as well just save themselves some cash and watch the original again.