Technically there aren't actually any "remote" things at all1 in your Git repo, there are just local names that should correspond to the names on another, different repo. The ones named origin/whatever will initially match up with those on the repo you cloned-from:
git clone ssh://some.where.out.there/some/path/to/repo # or git://some.where...
makes a local copy of the other repo. Along the way it notes all the branches that were there, and the commits those refer-to, and sticks those into your local repo under the names refs/remotes/origin/.
Depending on how long you go before you git fetch or equivalent to update "my copy of what's some.where.out.there", they may change their branches around, create new ones, and delete some. When you do your git fetch (or git pull which is really fetch plus merge), your repo will make copies of their new work and change all the refs/remotes/origin/<name> entries as needed. It's that moment of fetching that makes everything match up (well, that, and the initial clone, and some cases of pushing too—basically whenever Git gets a chance to check—but see caveat below).
Git normally has you refer to your own refs/heads/<name> as just <name>, and the remote ones as origin/<name>, and it all just works because it's obvious which one is which. It's sometimes possible to create your own branch names that make it not obvious, but don't worry about that until it happens. :-) Just give Git the shortest name that makes it obvious, and it will go from there: origin/master is "where master was over there last time I checked", and master is "where master is over here based on what I have been doing". Run git fetch to update Git on "where master is over there" as needed.
Caveat: in versions of Git older than 1.8.4, git fetch has some modes that don't update "where master is over there" (more precisely, modes that don't update any remote-tracking branches). Running git fetch origin, or git fetch --all, or even just git fetch, does update. Running git fetch origin master doesn't. Unfortunately, this "doesn't update" mode is triggered by ordinary git pull. (This is mainly just a minor annoyance and is fixed in Git 1.8.4 and later.)
1Well, there is one thing that is called a "remote". But that's also local! The name origin is the thing Git calls "a remote". It's basically just a short name for the URL you used when you did the clone. It's also where the origin in origin/master comes from. The name origin/master is called a remote-tracking branch, which sometimes gets shortened to "remote branch", especially in older or more informal documentation.