I've tried to download flash player several times onto my iMac, but still can't run any videos.

You can check here: http://www.adobe.com/products/flash/about/ to see which version you should install for your Mac and OS.

(If you are running Mavericks: After years of fighting malware and exploits facilitated through Adobe's Flash Player, the company is taking advantage of Apple's new App Sandbox feature to restrict malicious code from running outside of Safari in OS X Mavericks.)

You should first uninstall any previous version of Flash Player, using the uninstaller from here (make sure you use the correct one!):

http://kb2.adobe.com/cps/909/cpsid_90906.html

and also that you follow the instructions closely, such as closing ALL applications (including Safari) first before installing. You must also carry out a permission repair after installing anything from Adobe.

After installing, reboot your Mac and relaunch Safari, then in Safari Preferences/Security enable ‘Allow Plugins’. If you are running 10.6.8 or later:

When you have installed the latest version of Flash, relaunch Safari and test.

If you're getting a "blocked plug-in" error, then in System Preferences… ▹ Flash Player ▹ Advanced

click Check Now. Quit and relaunch your browser.

You can also try these illustrated instructions from C F McBlob to perform a full "clean install", which will resolve the "Blocked Plug-in" message when trying to update via the GUI updater from Adobe.

Use the FULL installer for 12.0.0.44: Flash Player 12 (Mac OS X)

And the instructons are here: Snow Leopard Clean Install.pdf

I cannot find Flash Player anywhere on my iMac. I have looked under Applications, Devices, Documents, and even Utilities. Nothing.

That's because it's not in any of those places. It is a "plug-in."

It's located in the root "Library" folder in a folder named, "Internet Plug-Ins," and the file is named, "Flash Player.plugin."