Check-in by definition is a process of registering the guest arrival at a hotel.
The guest may be asked to present a hotel voucher or booking confirmation and a piece of identification such as a national id, a passport or a driving license in the US. Sometimes a deposit needs to be authorized on a credit card or paid in cash.
After the formalities, the guest receives a room key and usually a short introduction on how to get to the room, how to get around the hotel and at what time breakfast is served.
During a check-in, a receptionist makes sure that all data regarding the guest stay are correct and in certain countries needs to proceed with registering the guest with the proper authority and pay a tourist tax.
At the time of check in the room has been usually already assigned to the guest by the reservation department, but if not, then it is up to the receptionist.
Depending on the hotel policy, it might be up to the receptionist to upgrade the room for the guest at this time.