Circumstances change people and circumstances should change characters too. This doesn’t always have to be a huge shift such as going from selfish to selfless, it could be as seemingly small as getting a different perspective on life. Whatever it is when characters begin a story they are rarely exactly the same at the end of it; they may be stronger or weaker, richer or poorer, optimistic or cynical. It may even be as simple as they learned a lesson and won’t make the mistake again.
The problem with the idea of a character changing over the course of a story is that writers can become worried that their characters haven’t changed enough. Occasionally we may even have a character who doesn’t change at all and simply bounces back at the end, like Morse in the books by Colin Dexter, he solves the case but the case doesn’t change him. If a character isn’t going to change then they need to begin as an interesting character with and interesting story and, probably, a degree of stubbornness. Readers loved the character of Morse and didn’t want him to change so it worked that he didn’t. However, this can be difficult to pull off because it can lead to a risk of repetition or can appear a flat if not done well.
In most cases the character will change through the story, even if it’s only a little bit. This is usually called the character arc which can be part of the problem writers encounter because ‘arc’ implies a conspicuous trajectory. However, people can change without realising and therefore so can characters. The change might not even be an obvious one because the change is the character realising something about themselves that they didn’t know before. For instance they might not think they are brave but by the end of the story they have proved that they are. It might even be a case that while they can’t see they are brave their actions show the reader that they are while they protest they aren’t.
Normally though when people think of a character arc they think of a situation where the character radically changes. An example would be in The Wizard of Oz where the characters go on a quest to find traits they wished they had and by the time they get to the end they have found them. You might argue though that, for instance, the Cowardly Lion was brave all along and simply didn’t realise it until he had an opportunity to be brave.
Sometimes finding your character arc can be as simple as getting to the end of the first draft and thinking ‘did the character know this at the start?’ If not then you have a character arc, even if it isn’t a conspicuous one.