The front desk is where the people who can check you in work at. They might be at a hotel to give you your key, or they might be at an office to check you in at an appointment.

Ask a clerk at the desk to check it for you!

You check it out at the front desk.

The check-in desk at any airport is always the first point of contact after entering the terminal. It's the first desk you see upon entering the building. This is the same for the airport, however it would depend on the airline used.

Going through a program on paper to check whether it has errors.