First, there is only one Catholic Church.
For a hospital to be tied with the Church there are a couple of things: 1) it must have some administrative functions carried out by either a Catholic diocese or a religious order in good standing with the Church, and/or 2) it is administered by an non-religious organization (read: an organization that isn't a group of monks or nuns) that sees its role as an apostolate of the Church (and is usually based in a certain diocese and subject to the authority of that diocesan bishop).
A Catholic Hospital is a non-profit hospital that exists solely to provide healthcare to a community. Most operate from the fees charged to patients, along with some assistance from the local diocese. As a Catholic Hospital, there is a Catholic chapel on premises, in which the Catholic chaplain (generally a priest) will say Mass daily. In order to be called a "Catholic" hospital, it must adhere to the moral teachings of the Catholic Church, as decided by the bishop in whose jurisdiction the hospital lies.