It's quite easy to find installation instructions for most Ubuntu versions but it's nice to check they still apply to the most recent release (15.04 at the time writing). I tested this yesterday and it still works for me. My thanks to Hongru Hou (link at the end of the article for full details).
Install Apache2 (if not installed):-
sudo apt-get install apache2
sudo apt-get update
Test Apache2 web server is working:-
sudo /etc/init.d/apache2 start
sudo /etc/init.d/apache2 stop
sudo /etc/init.d/apache2 restart
Install Java 7 using the 'Webupd8team' repository:-
sudo add-apt-repository ppa:webupd8team/java
then:-
sudo apt-get install oracle-java7-installer
(takes about 10 minutes to download)
Install Tomcat7 (because it is GeoServer Web archive this replaces the missing Java Servlet of this edition)
sudo apt-get install tomcat7
then in order for tomcat7 to find JDK we need to open and amend one line:-
sudo vi /etc/default/tomcat7
JAVA_HOME=/usr/lib/jvm/java-7-oracle
restart tomcat7:-
sudo /etc/init.d/tomcat7 restart
Download GeoServer 2.4.0 archive from:-
http://sourceforge.net/projects/geoserver/files/GeoServer/2.4.0/geoserver-2.4.0-war.zip/download
Extract the .war file somewhere convenient then:-
gksudo nautilus (password required)
and manually copy/paste the .war file into:-
/var/lib/tomcat7/webapps/
then restart apache2 again:-
/etc/init.d/apache2 restart
Now you should get your GeoServer login page:-
http://localhost:8080/geoserver
with some nice demo data included to get you started.
My thanks to Hongru Hou for his original blog:-
http://hongru.hou.io/blog/2013/01/07/How-to-install-GeoServer-on-Ubuntu-12.04/