The air conditioning doesn't work
A very common, slightly more informal way to say 'it doesn't work'. 'Servir' is frequently used for machines.
Use this when the main problem is a lack of cold air. It means 'It doesn't cool'.
This means 'The air conditioning is damaged'. It implies a more serious, physical problem.
Use this in regions like Mexico where 'el clima' is a common synonym for 'air conditioning'.
Excuse me, the air conditioning doesn't work in my room.
It's very hot in here because the air conditioning doesn't work.
I called the front desk. The air conditioning isn't cooling at all.
My car's AC is damaged, I need a mechanic.
The standard term is 'el aire acondicionado'. The verb 'funcionar' (to function) is the most technically correct verb for machines.
While 'aire acondicionado' is understood, it is very common to hear 'el clima' used as a synonym for the A/C unit. For example, 'El clima no funciona'.
They also use 'el aire acondicionado'. To say something is broken, besides 'no funciona', you might also hear 'está estropeado'.
'El aire acondicionado' is the standard term. To say it's broken, you will commonly hear 'está roto' (it's broken) in addition to 'no funciona'.
Using 'trabajar' instead of 'funcionar'. — Don't say 'El aire no trabaja'. 'Trabajar' means 'to work' for people. For machines, the correct verb is 'funcionar' (to function) or 'servir' (to work/serve a purpose).
Using the wrong gender for 'aire'. — Don't say 'la aire'. 'Aire' is a masculine noun, so it's always 'el aire acondicionado'. The same is true for 'el clima'.
Directly translating 'it doesn't make cold'. — Saying 'no hace frío' is awkward. The natural way to say the A/C isn't cooling is 'no enfría' (it doesn't cool), from the verb 'enfriar'.