Broadly speaking, affiliations describe groupings of Amish churches of similar practice, identity, and lifestyle. The best discussion of Amish affiliations I’ve come across is in Chapter 8 of The Amish by Donald Kraybill, Karen Johnson-Weiner, and Steven Nolt.
The Kauffman Amish Mennonites, also called Sleeping Preacher Churches or Tampico Amish Mennonite Churches, are a Plain, car-driving branch of the Amish Mennonites whose tradition goes back to John D. Kauffman (1847-1913) who preached while being in a state of trance and who was seen as a "sleeping preacher".
The Beachys favored a milder discipline for members whose only offense was transferring membership to other Anabaptist churches, specifically the conservative Amish Mennonite congregation that broke from Moses Beachy's congregation (then not under Beachy's leadership) in 1895.
The Amish are a Christian denomination, originally a subsect of the Mennonites who were in turn a subsect of the Anabaptists (or "re-baptizers," because they re-baptized adult converts who had been baptized as infants in the Catholic church or in very early Protestant churches that still practiced infant baptism; children born into the Amish tradition then and now are not baptized as infants, but only after age 16 or more when they make their own profession of faith).
The Old Order Amish is the concept many outsiders have when they think of "Amish". In 1990, Old Order Amish settlements existed in 20 states in the United States and in one province in Canada. Membership was estimated at over 80,000 in almost 900 church districts.
Subgroups of Amish developed over the years, as Amish churches have divided many times over doctrinal disputes. The 'Old Order' Amish, a conservative faction that withdrew from fellowship with the wider body of Amish in the 1860s, are those that have most emphasized traditional practices and beliefs. There are many different subgroups of Amish with most belonging, in ascending order of conservatism, to the Beachy Amish, New Order, Old Order, or Swartzentruber Amish groups.