I keep hearing the terms "character development" and "characterisation", but I'm still not sure what they mean? How does one "develop" a character?
Character development can refer to either the task of sitting down and creating a character (working out their appearance, history, mannerisms, and so on), or it can refer to the change a character undergoes during the course of a story.
In the first instance, the idea is to create a fictional person, complete with flaws and weaknesses, history, mannerisms, hopes, fears, someone that is often even more real than people you meet in real life. This is (generally) a person you will use in your story, whether as a protagonist or antagonist. It's like you getting to know someone so well that you know absolutely everything about them. Just how much you need to know depends on how important the character is to the story, but generally speaking, the more you know about the character, the better able you'll be able to write about them in relation to your story.
In the second instance, character development describes the change an individual undergoes through the course of a story as a result of the conflict or conflicts that person encounters. Think of the character Scrooge from A Christmas Carol, who begins as a crotchety, tight-fisted, greedy old man, but over the course of the story, various "conflicts" force him to change his ways so that, by the end, he becomes a better man, generous and kind.