The compressor is the part of your home cooling system that allows it to actually cool. Without the compressor, the refrigerant in the AC wouldn’t be able to change between liquid and gas form, releasing and absorbing heat as it does so. It also wouldn’t be put under pressure so it would circulate. If you don’t have a working compressor in an AC, you don’t have an AC.
There are other vital parts of an air conditioning system, but a failed compressor is perhaps the most expensive to fix. So much so that people often find it more cost-effective to replace the air conditioner entirely rather than pay to replace only the compressor.