When downloading an rar, what exactly does it mean when one of the
components shows up as, "file header broken?" Is there a way to repair
it?

Try downloading archive again, preferrably with some download manager
to make sure you get exact copy of remote file (downloads can be
broken on lower quality connections) and make sure you have the latest
version of RAR (v3.42 for Windows is the lates and can be found at
http://www.rarsoft.com/rar/wrar342.exe) because it may be that newer
versions support additional header information.

If that doesn't help, it's most probably because remote file is broken
or damaged in some way.

It means your file is corrupted (you probably downloaded incomplete or
with a little diference in bytes when it was compressed).

Try to re-donwload the rar file; if it dont work try the repair
utility in Winrar. WinRAR is able to repair only RAR and ZIP archives.
WinRAR tries to detect the type of the archive automatically using
archive data and extension, but user also may set an archive type
manually.

The source archive is not changed and the result of the operation is
stored into a new archive, which is named fixed.arcname.rar or
rebuilt.arcname.rar, where 'arcname' is the original archive name.

Chances of a successful repair of RAR archive are significantly
higher, if it has a recovery record.

Note that a solid archive without a recovery record cannot usually be
successfully repaired.

If you want to know more of Winrar check: 
http://www.techsono.com/faq/rar.html

Google search:
repair rar files

I hope it helps.
Phossil