You enable this with the Apache userdir module. You can enable it by doing this:
$ cd /etc/apache2/mods-enabled
$ sudo ln -s /etc/apache2/mods-available/userdir.conf .
$ sudo ln -s /etc/apache2/mods-available/userdir.load .
$ sudo /etc/init.d/apache2 restart
Now create a directory called public_html in your home directory and put your files in there. Make sure it is readable by everyone so the web server process can see it:
$ chmod -R 777 ~/public_html
Then you can access your files by going to http://localhost/~username where username is whatever your Ubuntu username is (type "whoami" in a terminal if you're not sure).
If you want to do this but don't want to have to have the ~username in the URL you can edit /etc/apache2/sites-available/000-default and change all instances of /var/www to /home/username/public_html, then you would see your HTML files by going to http://localhost/