If you haven't already i would suggest installing SUPEE-8788. This removes the flash uploader from the admin meaning that it uses the default html functionality instead. Most browsers block flash by default and you have to manually enable it for use on sites. This is to prevent vulnerabilities in flash.
Once the patch is applied, clear caches, log out/back into the admin and try again and you should be good to go.
Worth a read before applying: Possible Issues with SUPEE-8788
Edit after more information provided.
If you've just updated Magento (Which you should have put in your original question) then,
- Make sure you've cleared all caches, logged out of the admin and back in again
- Do you have the Creare SEO Extension? If so there is an incompatibility with it and SUPEE-8788 but there is a fix available for it here: https://github.com/adampmoss/CreareSEO/pull/78/files