To work with branches, the procedure is as follows:
If you do not have the branch down:
git clone -b nombre-rama https://github.com/nombre-repositorio
If you already have the folder with the branch down:
git pull
(you only need to pull because the .git already knows that you are working with the branch)
You make all the changes you want.
To upload it simply do:
git add . #puedes cambiar el '.' por el/los fichero/s concreto que quieras.
git commit -m "Descripción de tus cambios"
git push
Keep in mind that between pull and push, if your partner makes a change in the branch and upload it, it will conflict with the last version you downloaded and will not let you upload it. Then you would have to do a pull to copy the last changes of your partner again.