While Fork (in Software Development) traditionally meant starting an independent development based on a copy of an existing project, a git "fork", as introduced by GitHub, is a mean to collaborate on an existing repository. This same concept has been extended to other hosting services such as Bitbucket and GitLab.
Since you don't have the right to push back to a repo you don't own, you can fork it (copy on the server side: it is almost, but not quite, the same as a clone done on the server), getting a copy of a git repo that you own, i.e., to which you can push back.
Then, you can make a pull request, signaling to the maintainer of the original repo that you wish your contribution to be merged back into the original repo. Note that the precise mechanisms, and internal details, of pull requests vary from one hosting provider to another.