The air conditioning compressor is a pump that compresses refrigerant gas. The compression of the gas causes it to increase in pressure and temperature. The high pressure forces the refrigerant to circulate through the AC system. Most compressors are driven by an AC belt.
The most common failure of the AC compressor is that the compressor piston will become worn. One this happens, then the compressor will need replacing.
Most manufacturer factory warranties will cover an air conditioning compressor under the bumper-to-bumper coverage. Once the bumper-to-bumper coverage expires, there are extended car warranties available that will extend coverage of the air conditioning compressor.