In /etc/default folder there is grub file. Often settings in this file are used to alter boot options. In this case, particularly we're interested in this line : GRUB_CMDLINE_LINUX_DEFAULT= . To boot into console mode , there hast to be "text" written after equals sign, and in case you want gui boot , it would be "quiet splash" . To have ubuntu booting with startup messages displayed, it would be "splash". It appears that in your case you need to change splash to quiet splash
I'm booting with GUI enabled. Here's how those lines in my grub file look like:
GRUB_DEFAULT=0
#GRUB_HIDDEN_TIMEOUT=0
GRUB_HIDDEN_TIMEOUT_QUIET=true
GRUB_TIMEOUT=5
GRUB_DISTRIBUTOR=`lsb_release -i -s 2> /dev/null || echo Debian`
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash"
GRUB_CMDLINE_LINUX=""
Look at this file , and see what it says. (You could use this command just to view cat /etc/default/grub). Before editing, make a backup copy of the file, sudo cp grub grub.backup . To edit, use this command sudo nano /etc/default/grub. You didn't specify how much you are familiar with command line in your original question, so just in case you're not, nano is a command line text editor. When you're done editing, just press Ctrl+X, then Y, then enter. Run sudo update-grub . After that reboot, see what happens.
Sources: askubuntu.com and askubuntu.com ; mainly for your question you should refer to the second one.
Please let me know if this helps. Also, other fellow ubuntu users, please post comments or suggested improvements to my answer