How does the car AC work? Is it connected to the engine or the battery?

Car AC consists of many components, some parts need power as follows:

1. Compressor: connected to the engine via serpentine belt

2. Condenser Fan: located in front of condenser, connected to battery, and controlled by switches and relays.

3. Cabin air blower: located behind your dashboard, connected to battery, controlled by switches...

4. Related sensors: like outdoor temperature sensor, cabin air temperature. They are part of the control system and connected to battery.

Once your start your engine, the belt will also drive and energize the alternator which keeps the battery charged. So, in a way, engine is also charging the battery.

Other components that don't need power are for example: Condenser, evaporator, valves, filters ... and those are not covered in your question.

How AC works, this is very big question, but to make it simple:

1. Compressor compresses the refrigerant gas and increases temperature and pressure

2. Condenser + Fan: located at the front of car just behind the front bumper. It cools down and condenses the gas to liquid by exchanging heat with the atmosphere. still high pressure (subcooled)

3. Expansion Valve: located near evaporator (behind dashboard). It "sprays and regulate" the liquid into the evaporator, lowering pressure and then the temperature.

4. Evaporator + blower: use the heat in the circulated air to evaporate and superheat the "regulated sprays" of liquid into gas, happily, the circulated air will get cooler in the process. So this is to exchange heat between cabin hot air and cold refrigerant tubes / fins (evaporator)

So blower will circulate the air and you feel the cold air coming out of the vents in your car.

5. The circle continues as the evaporated gas (superheated) returns back to compressor (by suction pressure)

Notes:

Fan is to increase effectiveness of condenser, also to move air across condenser when you are at very low speeds or stopped like traffic signal / traffic jam.

Blower is to circulate air through evaporator and cabin, so the process will success to cool the air and evaporate the refrigerant effectively.

Have a nice day.

It's connected to the battery 

As far as Ac of the car is concerned it's directly connected with main engine of the car. It needs both power and fuel whose only source is engine.