How should I uninstall/remove Adobe Flash Player in Ubuntu?

In case you installed it using Apt, run this to find the package name:

dpkg --get-selections | grep -v deinstall | grep flash | awk '{ print $1 }'

Then you can uninstall the package by

sudo apt-get purge package-name

In case you copied the plugin so to Firefox plugins, simply delete it from there.

Open synaptic package manager -> find flashplugin-installer -> Mark for Complete Removal -> Apply.

In reply to Elder's comment: "Removing the installer package won't remove the plugin", I just wanted to clarify that the pre-processing scripts of flashplugin-installer package, does remove the plugin, as long as you take care of marking it for Complete Removal or use "purge" action, as both the answers recommends.

If you want to check, just locate the plugin on your filesystem:

$ ll /usr/lib/mozilla/plugins/
flashplugin-alternative.so -> /etc/alternatives/mozilla-flashplugin

$ ll /etc/alternatives/mozilla-flashplugin
/etc/alternatives/mozilla-flashplugin -> /usr/lib/flashplugin-installer/libflashplayer.so

$ ll /usr/lib/flashplugin-installer/libflashplayer.so
/usr/lib/flashplugin-installer/libflashplayer.so

Now remove the flash-installer package with:

apt-get purge flashplugin-installer

and check that the plugin itself has been removed:

$ ll /usr/lib/mozilla/plugins/
No such file or directory

$ ll /etc/alternatives/mozilla-flashplugin
No such file or directory

$ ll /usr/lib/flashplugin-installer/libflashplayer.so
No such file or directory