A matinee is an afternoon show, which usually falls between noon and 5 p.m. An adult matinee can sometimes mean the distinction between the price of a ticket for an adult and the price of a ticket for a child. Matinees are often a few dollars cheaper than the regular evening prices, so when it's referred to as an "adult matinee" that refers to the price an adult will pay for an afternoon viewing.
Some independent theaters will hold adult matinees that are intended for adults only so that they can view a film without the distraction of children. Large movie chains don't have adult matinees in this sense.
Most live theater shows are also referring to ticket price when it comes to adult matinee. Theaters that hold live performances will also usually have cheaper tickets for the afternoon shows.
A lot of people enjoy matinees because they are usually less crowded and less expensive. Weekend matinees are naturally more crowded than during the week. Depending on the theater, you may have the experience of sharing the theater will very few people, or possibly no one.