These words mean an arrangement to see someone.
You make anappointmentwith a professional person, for example a doctor, lawyer, or hairdresser.
I made a dentist's appointment for Monday morning.
To refer to an appointment to eat in a restaurant or stay in a hotel, you can usereservationor, in UK English,booking.
I have a reservation for 8 pm.
UKI have a booking in the name of Smith.
Acommitmentis something that you need to do or deal with that takes your time. It is used more in writing than it is in speech.
I have no commitments tomorrow, so let’s meet up in the morning.
Adateis a particular kind of appointment. It is a social meeting with someone you are romantically interested in.
We went out on a date last week.