The Teip is an old organisational principle of Vainakh society that survives into the present day. Modern society is split into 150-160 Teips, each of which traces its origins to a common ancestor and a specific geographic location. So when two people say they’re from the same Teip, this means that if you go back in their family tree far enough you will find a shared ancestor.
Teips would form alliances in order to defend themselves against attack and these alliances are called Tukkhums. Although the boundaries between Tukkhum and Teip have become blurred, there are broadly speaking 9 Tukkhums in Chechnya today.