How to enable boot messages to be printed on screen during boot up?

You'd need to remove the kernel boot parameters quiet and splash from the linux line in GRUB:

1. Start your system and wait for the GRUB menu to show (if you don't see a GRUB menu, press and hold the left Shift key right after starting the system).
2. Now highlight the kernel you want to use, and press the e key. You should be able to see and edit the commands associated with the highlighted kernel.
3. Go down to the line starting with linux and remove the parameters quiet and splash.
4. Now press Ctrl + x to boot.

To make this change permanent:

1. From a terminal (or after pressing Alt + F2) run:

   gksudo gedit /etc/default/grub
   

   and enter your password.

2. Find the line starting with GRUB_CMDLINE_LINUX_DEFAULT and remove the parameters quiet and splash.

3. Save the file and close the editor.

4. Finally, start a terminal and run:

   sudo update-grub
   

   to update GRUB's configuration file.

Edit you bootloader kernel command-line and remove quiet argument. You may also want to remove splash argument to disable graphical animation during boot to be able to see the console with messages.

In more recent releases, including 12.04 and 12.10 it is also necessary to either delete, or change the GFXMODE line to "text" as well as removing "quiet" and "splash". (See the other answers.)

When making the changes permanent find the commented line "#GRUB_GFXMODE=640x480" and both remove the beginning "#" so it is no longer a comment and set its value to "text".

With the latter change the menu will be white on black in a larger text font and therefore hold fewer entries, but the messages should be visible once an item is selected for booting.