In general, umbilicals are provided from the launch pad to the vehicle for any services that need to be provided after the vehicle is installed on the launch mount, and to remove hazardous gases from the vicinity of the vehicle.
Consider that vehicles can sit on the pad for long periods if problems occur during the countdown. Many different consumables may need to be replenished. Use of ground supplied power and cooling conserves vehicle resources for flight.
Details vary, but will be similar for all vehicles.
For the Space Shuttle, the top umbilical arm (the "beanie cap") connected to the liquid oxygen tank vent at the tip of the External Tank, and carried oxygen vapors away from the vehicle.
The next arm down, which passes on the other side of the solid rocket booster in this picture, is the External Tank Hydrogen Vent and Intertank Access Arm...which does exactly as its name suggests.
The next arm down is the Crew Access Arm.
Most interfacing services on the Shuttle were provided by the Tail Service Masts, the gray tombstone-shaped structures to the left and right of the Orbiter tail. They supplied cold coolant, purge gas, liquid oxygen, liquid hydrogen, gaseous nitrogen and gaseous helium, and returned warm coolant, recirculated liquid oxygen and liquid hydrogen to the ground facilities. Sense line connections for the leak detection system were provided. They also supplied connections for electrical power, communications, and data.
Here is a schematic of one of the T-0 umbilicals showing the various connectors and a list of the services provided through them. I haven't found a list for the other umbilical but it was very similar.