A standard air conditioner has two modes of operation. In technical terms, they are called "on" and "off". When an air conditioner is "on", it will remove heat from the air and blow out air that is below the ambient temperature. When an air conditioner is "off", it will not do anything. If the fan is blowing when the air conditioner is "off" (possible in some home a/c setups and in pretty much all automotive applications), the air that is being exhausted will be the same temperature as ambient. An air conditioner that is "off" is indistinguishable from having no air conditioner at all as far as its effect on ambient temperature is concerned.
Now, a thermostat is a device that operates as a fancy on/off switch. You set a temperature (hence the whole "thermo" part of the name, which is derived from a Greek word meaning "heat"), and then the thermostat monitors the ambient temperature, and if it is above that temperature, it turns the air conditioner "on", whereas if it is below that temperature, it turns the air conditioner "off". (Note that for the operation of a heater, "on" and "off" are reversed.) Yes, there are some minor caveats around the above description in that there is general +/-1 or 2 degrees of hysteresis built into the system in order to prevent on/off oscillations that can be damaging to an air conditioner's compressor, but to first order, if it is hotter than the setting, it will turn the air conditioner "on", and it will keep it "on" until is is colder than the setting, at which point it will turn the air conditioner "off".
So, let's say you enter into an environment in which the air conditioner has been off for a while (like a parked car or a house after returning from vacation), and the environment is hotter than you like. You set the thermostat to the temperature you desire and then enable the air conditioner. You notice that it is taking some time to cool off.