As an airline customer service agent, you would usually work for an airline, or for a ground services agent on behalf of an airline. Your job would include:
dealing with passenger enquiries about flight departures and arrivals
checking passengers in
giving seat numbers
providing boarding passes and luggage labels
telling passengers about luggage restrictions
weighing baggage and collecting any excess weight charges
taking care of people with special needs, and unaccompanied children
calming and reassuring nervous passengers.
You may sometimes help passengers through immigration and customs, or escort passengers who have night flight connections. You could also specialise in different areas of airport work, such as computer control.