First we need a way to send an e-mail. This package provides the "mail" command.

apt-get install mailutils

As root, create the file /etc/ssh/sshrc with the following content:

exec 1>&2
env | mail -s "LOGIN $USER@$(hostname) from ${SSH_CONNECTION%% *}" root

From now on the user "root" will receive an e-mail with the user who logged in and their IP-address.

This is only a practical way to keep track of users logging in to a shell. It's not a way to audit every login because it's easy to bypass, for example by logging in without starting a shell. For auditing it's better to watch the logs.