Adobe Flash Player won't install properly when it thinks that you've already got it installed, even if that installation is actually corrupt.
Go to your list of installed programs (via the Control Panel) and uninstall any references to 'Adobe Flash'. (There might be just one but there could be several).
RESTART YOUR COMPUTER. (That's in capitals because it's important!).
The go back to
https://get.adobe.com/flashplayer/ and install Flash again.
You should note that many browsers now have a default setting which blocks Flash from running (because it can significantly reduce your browser's performance and, unless it's kept fully up to date, present a security risk to your computer). So when you go to a site which uses Flash you might have to click on 'Enable Flash', and then refresh the page, before getting it to work.