[I’m going to eschew the classical conflicts. They may be useful in English class or in literary discussions, but we need to look at them from a writer’s perspective.]
In simplistic terms, a conflict is what the main issue of a given character; it’s what he’s trying to solve or deal with. As far as the story is concerned, all conflicts should tie into the theme.
You are going to have various conflicts going on. Obviously, you will have some sort of exterior conflict that overarches all other conflicts; individual conflicts need to be defined in terms of the big exterior conflict. War, or other competition, is popular for this purpose because, well, they are easy to get everyone into. Consider war for a moment: It involves all the characters, and aligns them into two or more teams. It becomes really easy to determine who is allied with who, and it helps simplify work for the writer.
Same with competitions, however, competitions allow you to break down the big sides into individual teams, and complicate things a bit. After all, you can go crazy defining allies and competitors, even aligning them into meta-teams, and it can get really weird really quick. For example, you can have a lot of teams that come from various countries/cities/villages, and the teams from those villages are aligned together. You can have a team from one country ally with a team from another country in order to help nail a third team from either a completely different country, or from one of the two countries already mentioned. Even with just four teams from a country, and three countries, you can have endless possible conflicts.
Another possibility is to have something that forces everyone to work together, like a natural disaster, a bigger enemy, or even just something (like getting caught on a ship in a storm) that forces them together. You can still have one (or more) sides working to use the emergency to off one of the sides, or even weaken another side for an easy kill afterwards, but they need to work together for the duration of the emergency.
However, each character needs to have at least one defining conflict, preferably an interior one. For example, are they having a crisis of personal faith? Problems with (annoying/missing/dead) parents? Trying to learn how to use new abilities? Not sure if they can handle the stress of the situation? Have an ability that can kill them if used, and thus afraid to use it? Heck, afraid of losing their humanity? How about trying to fall in love? Or even not sure if they belong to the group in question?
You’ll find that most inner conflicts are faith or confidence related. Most people are not sure of themselves on some level, and writers are definitely in that group, and so it’s easy to relate to. You need to realize that a conflict is created by wanting something that they lack; figure out what the missing element is, and that’s your conflict.
And bear in mind that even your antagonists need conflicts as well. Sure, Dread Overlord Deathmate wants to take over the multiverse, but why? If he wants to destroy it, don’t just leave it at that; why does he want to destroy it? Maybe he fell in love with someone long ago, and wants to commit suicide so he can forget her, but can’t bring himself to commit suicide, but if he tries to destroy the multiverse, someone may succeed at killing him (thus committing suicide by proxy). Obviously, overcoming that grief or getting killed is his conflict. Lesser villains obviously have smaller conflicts.
Conflicts tend to make your character more three-dimensional; after all, they may have really cool powers and skills, and even a function, but they need the conflict to make them even more human. They also help you write them, as you now have a way to link them to the plot (a conflict makes their involvement necessary, so that they can resolve their conflict), as well as something to do with that character (if you ever get writer’s block, you can just look at any unresolved conflicts, and look at ways to at least partially resolve them).
In short, conflicts not only make your life easier, but they make sure that you keep readers...