Formally called “The Society of Jesus,” the Jesuits (“JEZ-sue-its”) is a subdivision of the Catholic Church. It was founded in 1540 in Spain by St. Ignatius of Loyola.
The Society of Jesus is only open to men. There is a formal membership process which can last for many years. Once accepted into the order, Jesuits serve in missionary or other service posts around the world. They are directed by the Superior General who reports to the Pope in Rome.
There are approximately 14,000 members around the world. Jesuits have their own classification of priest, which is different than a mainstream Catholic priest. Jesuits who are not priest are called “brothers,” and they are free to marry.
Some Catholics consider themselves “lay Jesuits,” without formal membership or application. Generally, they commit themselves to service and a simple lifestyle.
The Jesuits have a long history of education. There are 27 official Jesuit colleges in the United States (the nearest to me is Creighton, in Omaha), and 60+ high schools. This page at Xavier University explains a Jesuit education and how it differs from the norm, however, the differences are quite vague and subjective.