During the turnaround many vehicles support the aircraft on the gate. What are the name of those "vehicles"and their purpose?
The most common vehicle seen is the tug.
Tugs are multipurpose tractors that are used to tow around all sorts of things.
Several types of equipment are shown in this diagram:
The tow vehicle at the nose is more commonly referred to as a pushback tug. They come in different sizes. Here is an example:
The GPU is most often a unit that is towed behind a tug. source
Belt loaders usually can be driven on their own:
The wide loader is used to put cargo containers in the hold: source
The cabin cleaning truck and galley service truck often have a lift built in:
Luggage can be brought in baggage carts
or in cargo containers which are brought to the aircraft on dollies source
Another vehicle often seen is the people mover:
In general, you could refer to most of the vehicles and their attachments/trailers as ground support equipment (GSE)
The vehicle is usually called a ground tug.