Passenger transports allow you to take several people into your ship and ferry them to other locations. If they're satisfied with the speed of the move, they will reward you accordingly.
To obtain passengers, visit a passenger transport facility and type REQUEST.
To figure out where each passenger wants to go, talk to them. If, say, you're transporting a dirty man, you would type TALK TO DIRTY MAN. He will then present you with a menu of conversation topics, one of which should be "Where are you going?"
Dropping Passengers Off:
Dropping the passengers off is a simple matter of landing on the planet they wish to go or docking with a space station. They will automatically pay you (assuming you're in the control room) and leave the ship.
Should you decide you no longer wish to transport the passengers, you can type DISMISS <passenger name> to dismiss them from your ship. You must be landed to do this.