Permanently disable color in default terminal

In your .bashrc file, you'll find lines

# uncomment for a colored prompt, if the terminal has the capability; turned
# off by default to not distract the user: the focus in a terminal window
# should be on the output of commands, not on the prompt
force_color_prompt=yes

Put a # comment sign in front of force_color_prompt=yes line and restart terminal. After the change, the line should be

#force_color_prompt=yes

Edit: This will turn of the color from terminal. But if some programs has color support built-in in them and if you want to turn them off too, follow @Zanna's answer. Basically you need to comment out the lines like alias ls='ls --color=auto' etc.

If you have set a coloured prompt see @Anwar's answer

The colours from the output of commands such as ls are made by aliases.

To disable this, find and comment out (by inserting # at the start of the line) these lines in your ~/.bashrc

alias egrep='egrep --color=auto'
alias fgrep='fgrep --color=auto'
alias grep='grep --color=auto'
alias ls='ls --color=auto'

when done, source ~/.bashrc to get the immediate effect or just close the terminal and open a new one

A simple, elegant solution.

No code needed for "GNOME terminal": just follow these steps:

1. Run gnome-terminal.
2. Go to Terminal > Preferences.
3. Go to Profiles.
4. Select the profile you use, then click Clone.
5. Go to Colours.
6. Replace every color of the bottom row with #EEEEEC or the rightmost color you have set.
7. Replace every color of the top row with #300A24 (default GTK theme background color) or the one your theme has. If you have disabled the system theme, set them to the "Background colour" instead.

Now you are almost done, but I recommend you to do these steps:

1. Go to General
2. Change the profile name to something like <PROFILE> (monochrome), replacing <PROFILE> with your old profile name.

Regardless of the above steps, you also need to follow these in order to successfully create your new profile:

1. Close the Editing Profile window.
2. On the drop-down menu, select the new profile you just created.
3. Restart gnome-terminal.