“Cousin” is one of those words that people sometimes use in a variety of ways. It may mean first cousin (where two individuals have parents that were siblings), it may mean a more distant cousin relationship, it may be a cousin-by-marriage, someone who is related in an unknown way, or someone who may not even be related at all.
Genealogists are sometimes precise in their use of cousin relationships. Other people often are not and sometimes genealogists find it easier to use the word “cousin” instead of the precise term for the relationship.
Don’t assume the precise nature of the relationship when someone is referred to as a “cousin.” Research it to try and determine what it is.
And remember that there’s always the chance that the person’s not actually related at all. Your ancestor may have said on his immigration papers that he was coming to live with his “cousin” when in fact that “cousin” was simply a former neighbor who agreed to help him get settled and saying they were “cousins” made the paperwork flow a little easier.