I can't get any Flash content to load on Chrome. When I try to watch a YouTube video or something like that, it just says, "Couldn't load plug-in".
You need to update Chrome to version 24.0.1312.69 or later. This is a known issue which has been fixed in the said version. Run the following command in terminal:
Source: Google Chrome Forum Post
Try the Adobe Flash plugin, using the flashplugin-installer to get the Mozilla style NPAPI version (from the "nonfree" section of the Ubuntu repositories). It's frozen at version 11.2 but still has security updates and works fine with most sites. Plug chrome://plugins into the address bar and disable the libpepflashplayer.
Then try switching back to the Pepper Flash sometime to see if it works again.
Edit: My guess was that not all websites were compatible with the newest Flash, but it looks like it was just a bug (see other answer about upgrading Chrome). There were other issues with Pepper when it came to full screening video in a previous release, so it's not a bad idea to have both plugins handy for when you upgrade and everything breaks.