These instructions will install Pepper flash player for Chromium and it will also be updated automatically. Because Google ended support for Chrome on 32-bit Linux in March, 2016, Pepper flash player can only be installed for 64-bit versions of Chromium.
To install flash player in Chromium web browser search the Dash for Software & Updates and open the Software & Updates window. Click the Other Software tab in the Software & Updates window and put a check mark in the checkbox to the left of where it says: Canonical Partners.

Click the Close button to close the Software & Updates window.
Open the terminal and type:
sudo apt remove pepperflashplugin-nonfree # remove pepperflashplugin-nonfree if it's installed
sudo apt-get update
sudo apt-get install adobe-flashplugin
sudo apt install browser-plugin-freshplayer-pepperflash # 16.04 and later
Original answer
The more up-to-date Pepper Flash Player that can be installed together with Chromium Web Browser will run with fewer glitches (such as fullscreen doesn't work, buttons on video don't work, video hangs after pausing and then pressing the play button to resume, etc.) than Adobe Flash Player for many Flash videos.
You can install Chromium web browser together with the Pepper Flash Player in Ubuntu 14.04 from the Ubuntu Software Center. Open the Ubuntu Software Center and search for Chromium Web Browser and Pepper Flash Player - browser plugin and install them.
If a more recent version of Pepper Flash Player has been released and you want to update Pepper Flash Player to the latest version, you can do this from the terminal using the following command:
sudo update-pepperflashplugin-nonfree --install
Pepper Flash Player cannot be updated with the following command anymore on 32-bit OSs since Google ended support for Chrome on 32-bit Linux in March, 2016. If you are using Ubuntu 32-bit were using Ubuntu 32-bit before March, 2016, you can update Pepper Flash Player to the latest version from the terminal using the following command:
sudo update-pepperflashplugin-nonfree:i386 --install
Either of these two commands will download the latest version of Google Chrome (the file is larger than 40MB) and then update only the Pepper Flash Player with the more recent version of Pepper Flash Player that comes with Google Chrome without installing Google Chrome.
If a more recent version of Pepper Flash Player has been released and you want to update Pepper Flash Player to the latest version, you can do this from the terminal using the following command:
sudo update-pepperflashplugin-nonfree --install
If you are using Ubuntu 32-bit, you can update Pepper Flash Player to the latest version from the terminal using the following command:
sudo update-pepperflashplugin-nonfree:i386 --install
Either of these two commands will download the latest version of Google Chrome (the file is larger than 40MB) and then update only the Pepper Flash Player with the more recent version of Pepper Flash Player that is bundled with Google Chrome without installing Google Chrome.
If updating Pepper Flash Player to the latest version fails with an error like this:
ERROR: failed to retrieve status information from google : W: There is no public key available for the following key IDs:
1397BC53640DB551
It happens because Google changed its signing key. To fix it run the following commands:
gpg --keyserver pgp.mit.edu --recv-keys 1397BC53640DB551
gpg --export --armor 1397BC53640DB551 | sudo sh -c 'cat >> /usr/lib/pepperflashplugin-nonfree/pubkey-google.txt'
...and then try to update Pepper Flash Player to the latest version again.