Karriere, Berufslaufbahn, and Werdegang mean career.
Can we use any word alternatively or they need to be used depending on any specific context?
Karriere has an implication of success, (Berufs-)laufbahn doesn't. Also, Berufslaufbahn is obviously less of a foreign word (Fremdwort) than Karriere, which might make a difference in some contexts. For example, if you're a business that tries to recruit and would like to look cool, you will probably use Karriere.
When talking about the career paths of officials (Beamte), you're more likely to find "Laufbahn". (In the north German city of Kiel, there is a small pedestrian path with the name "Beamtenlaufbahn" that connects the city hall with the police headquarters. The name is a pun of course.)
Werdegang (lit.: "path of becoming") is mostly used retrospectively. For example, it could be used in a CV or bio, to describe how a person became who they are today. Or it could be used for a description of what steps a person would need to have done to be, say, a master carpenter.