The way NOT to do it is to go to the Adobe website and download it from there. That's just making life difficult for yourself. This is a general rule for all sorts of software on Linux; don't go straight to the website for the software. First you should look for it in the software repositories (Software Centre in Ubuntu). Only go to websites if you are installing obscure software that isn't in the repositories.
The package that installs Flash Player in Ubuntu is called flashplugin-installer.
There is also a package called ubuntu-restricted-extras, which installs Flash Player, Java and some other things which can't be installed by default for licensing reasons. In my instructions, I will use the flashplugin-installer package as an example, but you can type in ubuntu-restricted-extras instead if you like.
There are two normal methods for installing it:
Method 1.
a. Open the Ubuntu Software Centre.
b. Search for flashplugin-installer
c. Install the flash plugin from there.
Method 2.
a. Open a terminal.
b. Type sudo apt-get install flashplugin-installer
c. Press enter, and type in your password.
d. If it asks to install additional packages to satisfy dependencies, press y.
If Firefox was open while you installed the Flash plugin, close Firefox and re-open it, and then go to https://www.adobe.com/software/flash/about/ to make sure it was installed (if it's installed, you'll see an animation and it will tell you which version you have).
Alternatively, you can install Flash Player by clicking on the button at apps.ubuntu.com. This is equivalent to searching in the Software Centre and installing it from there. (The button just tells Ubuntu to look in its software repositories for the requested program.)
For ubuntu-restricted-extras (Flash, Java, other things) go to https://apps.ubuntu.com/cat/applications/ubuntu-restricted-extras/
For flashplugin-installer (just the Flash Player) go to https://apps.ubuntu.com/cat/applications/flashplugin-installer/
and click on the "available on the Software Centre" button to install the software.