You need to create a local branch that tracks a remote branch. The following command will create a local branch named daves_branch, tracking the remote branch origin/daves_branch. When you push your changes the remote branch will be updated.
For most versions of git:
git checkout --track origin/daves_branch
–track is shorthand for
git checkout -b [branch] [remotename]/[branch] where [remotename] isorigin in this case and [branch] is twice the same, daves_branch in this case.
For git 126.96.36.199 you needed this:
git checkout --track -b daves_branch origin/daves_branch
For git 188.8.131.52 and higher this is enough (might have started earlier but this is the earliest confirmation I could find quickly):
git checkout daves_branch