If your AC doesn’t turn off when room temperature reaches the temperature set on the thermostat, the problem is generally occurring either at the thermostat or in the electrical system that runs the outside condensing unit. You can use the AC’s circuit breaker to shut off the unit, but that should not be a permanent solution—circuit breakers are not designed to be used like light switches.
Fuses -- Anyone who has worked with electrical systems knows all about fuses and how they fail. They can burn out over time, may just be loose, or can blow out during an electrical storm or due to overload from another failed component. Of course, that's what they're supposed to do; they stop surges from going through and damaging the rest of the system. When a fuse fails, whatever system it was protecting will stop working.
Capacitors -- Capacitors help your HVAC motor start from a standstill. Over time they can weaken. As they weaken, they cause the motor to run hotter. This can shorten the life expectancy of your motor. Once a capacitor fails, the whole motor will stop working. You usually won't notice that a capacitor has weakened until your motor stops turning on.
Some of these components can be repaired or replaced by the homeowner, such as filters, fuses, and clogged up drain lines. Coils, compressors and the other components are best left to a professional. In some cases, you may have a system that is so old that parts are no longer available or else they aren't up to code. In this case you will have to consider replacing the entire system.