You need administrative permissions on the machine. If you don't have them, you'll need to talk to someone that does (IT, the machine owner, etc).
If you *do* have administrative access, it's typically because one of the files that need to be replaced is either currently in-use, or corrupt.
Grab the uninstaller. We'll just have you uninstall and re-install, as it eliminates a bunch of potential issues. Since you're already deep into it, we might as well be thorough...
Uninstall Flash Player - Windows:
Reboot, then run the uninstaller. Rebooting should resolve any issues with the files being in-use, and Windows will verify that the registry is in a good state at start-up. Unless you have something that launches immediately at startup that uses Flash, this should eliminate any potential problems.
If that doesn't work, you could try booting into Safe Mode to prevent any startup applications from launching.
If *that* still doesn't work, then it's probably a file on the hard disk that's corrupted. Go ahead and reboot again into normal mode.
Disk corruption can be a early sign that your hard disk is about to become a recently deceased hard disk. If you haven't backed up in a while and you have information that you would be sad to lose, this is a great time to make sure that your backup situation is in order. Once you've established that you have good backups of anything that you care about, check for and repair any problems with the filesystem.
Check your hard disk for errors - Windows 7:
After that, try manually removing Flash Player. If you get to a step that you can't complete because you're hitting permission problems or something similar, you've found the corrupted file.
Performing a Clean install of Flash Player on Windows:
I have really good luck with this utility called Unlocker, but we have no affiliation with them. It's just some freeware product I found years ago, and it gets the job done.
You'll probably want to read the editor's review on CNET before running the installer.
Once you have Unlocker installed, you can usually right-click on the offending file, choose Unlock, and then you'll be able to delete it.
Once you've completed the manual uninstallation, you should be good to run the installer again.