Matrilineal is a way of reckoning descent by mother's status. Which means that all of your legal and social status comes from whatever status your mother has.
In most Western societies, currently, descent is reckoned patrilinealy since babies get the father's last name, although since in most Western societies, people don't officially inherit social status, it could be said that we are neither patrilineal or matrilieal.
The most common example of matrilineal descent in the modern world is the Jewish people. A Jew is considered a Jew if they have a mother who is a Jew. However, Jews do adopt their Father's last name, not their mothers.
The issue of Jewish matrilineality brings up the issue of The Matrilineal descent of the Jewish people.