what do they call the place in america where people arrested get booked in

They usually call it booking or the booking area , and the place is often a jail or a police station where the intake happens.

In some cities, people may also say central booking , which is a local processing facility used after an arrest.

Simple wording

• Booking = the process of recording an arrested person’s information.

• Jail / police station = the place where booking usually happens.

• Central booking = the name some places use for the intake site.

Example

“After the arrest, they took him to central booking.” TL;DR: The common answer is booking , usually at a jail or police station.