Select the drive on your Desktop, and "Get Info" (Command - I) on it. Under "Format", make sure it's not MS-DOS FAT 32 or NTFS. If it is, then you need to reformat the drive to Mac OS Extended (Journaled).
Bear in mind that doing so will erase all data on that drive, but if things aren't backing up to it and there's nothing else on it, that's probably not an issue.
If your drive's format IS Mac OS Extended (Journaled), or some valid Mac format, then it's likely that you've either set Privacy exclusions incorrectly in the Time Machine control panel, or that Spotlight has taken a dump and hasn't indexed your drive correctly.
Tell us what you find.