Installation

apt install bash-completion

Enable

   1. Per User
   2. Globally

Enable per user

edit ~/.bashrc

if [ -f /etc/bash_completion ]; then
    . /etc/bash_completion
fi

OR

if ! shopt -oq posix; then
    if [ -f /usr/share/bash-completion/bash_completion ]; then
        . /usr/share/bash-completion/bash_completion
        elif [ -f /etc/bash_completion ]; then
            . /etc/bash_completion
        fi
    fi

To try it without logging out and back in, run:

. ~/.bashrc

Or open a new shell.
Then try to use tab-completion with apt.
That dot and space at the beginning (.) is the same as using the source keyword in bash, but is more portable.
If you want it to work when su'd into the root account, do the same thing in root's home directory (typically /root).

Enable globally

Do changes from (1) in /etc/bash.bashrc
To anyone who's wondering why this works, the . in front of /etc/bash_completion does not refer to the current directory, since it has spaces around it.
It instead makes the contents of the given file be evaluated in the currently running shell, instead of being executed in a new subshell. It is standardized here.
In Bash, this . can be replaced by the command source, but this is not standardized by POSIX and is less portable so I tend to steer people away from using it.
In this case, since it is specifically a program for extending bash, rather than something that needs to work in a bourne shell or ksh, you can feel free to substitute source for readability.
Incidentally, this behavior (not opening a sub-shell) is similar to the way DOS/Windows .BAT scripts work normally, changing the state of the shell they are run in.
This is why if you cd into a different path in a shell script, you won't be in that path when the script exits like you would be in a .BAT.