A patriarchal system is one run by men. Traditionally, most societies have been patriarchal, but it's now considered sexist and unfair.
The patriarchal way used to be pretty much the only way: countries and societies were run by men. The U.S. was very patriarchal, especially when women couldn't even vote. Over time, these things change, though probably not fast enough. These days, calling a country or business patriarchal is usually an insult — it implies that women aren't being treated equally.
