The phrase universal church is not found in the Bible, although the concept is. The universal church (or simply Church, as a proper noun) refers to all who follow Jesus (Matthew 16:18). As such, there is only one Church.
In some circles, the universal church is also called the catholic church, which is not to imply the Roman Catholic Church. However, it should be noted that the Roman Catholic Church is, by definition, part of the universal church.
Get Instant Access!
Sign up for Peter DeHaan's newsletter and get weekly info to guide you into a fresh, new faith--from a biblical perspective.