First fetch from the origin (safe, does not affect your local changes)
git fetch origin
Next compare your local branch with the branch on the origin (e.g. master)
git diff my_branch origin/master
If you are happy with the diff results (i.e. merging looks pain free) then merge remote into your local branch
git merge origin/master
You are now up to date and have merged in your local changes.
You can also use git pull to perform this task but then you are relying on an automated merge which is fine for most cases but in complex branching situations you might feel safer inspecting the changes first with git diff and merging manually.
If it all goes to hell then you can undo the merge with:
git reset --merge ORIG_HEAD