Here we go, yo, so what's the scenario?
You clone a Git repository from, say, GitHub, to dive into the source code. Eventually, you decide you would like to make some changes and submit a pull request. You need to switch your repo to point to your fork. Here's how.
List the current remote(s)
Here we see I cloned the Jenkins project, which was itself forked from Kohsuke's account.
$ git remote --verbose
origin https://github.com/jenkinsci/jenkins.git (fetch)
origin https://github.com/jenkinsci/jenkins.git (push)
upstream https://github.com/kohsuke/hudson.git (fetch)
upstream https://github.com/kohsuke/hudson.git (push)
Replace the existing origin with your fork
$ git remote rm origin
$ git remote add origin https://github.com/olivierdagenais/jenkins.git
Replace the existing upstream with the old origin
...if applicable, of course.
$ git remote rm upstream
$ git remote add upstream https://github.com/jenkinsci/jenkins.git
Double-check the remote(s)
$ git remote --verbose
origin https://github.com/olivierdagenais/jenkins.git (fetch)
origin https://github.com/olivierdagenais/jenkins.git (push)
upstream https://github.com/jenkinsci/jenkins.git (fetch)
upstream https://github.com/jenkinsci/jenkins.git (push)