Question: Forking on Github or Bitcket creates a repo's copy under your own account. 

Does this mean that forking is a server-side action? I was unable to find "fork command" inside Git.

If it's not a server-side action, what are the steps to fork a local Git repository with full history and tags?

Answer: 
Forking is just copying the whole repo. `git clone` is how you do it locally. Or you could literally just copy the entire repo.

See also [Are git forks actually git clones?](https://stackoverflow.com/questions/6286571/git-fork-is-git-clone)
There is a great summary of forking on [GitHub](https://help.github.com/articles/fork-a-repo/), but essentially you are creating a copy of their repo so you can make your own changes to it without affecting the source repo. 

This means you can build upon what someone else has done but alter it to your own ends.