Contactors are electromechanical switches that turn on the air conditioner compressor or electric heating elements. The critical ratings for a contactor are number of poles, coil voltage and contact amps. The coil voltage and contact amp ratings are typically marked on the tag of the contactor. The number of poles are actually the number of contacts or switches in the contactor.

In the picture below the contactor on the left is a single pole. It is manufactured using the same housing as the contactor on the right which is a double pole. You can see the contactor on the left has 1 set of contact points. The points on the left have been replaced in the housing with a bar that connects the top and bottom connections together.  The contactor on the right has 2 distinct sets of contact points so it is considered a double pole.

The amp ratings for a contactor are given as FLA for a motor load and RES for an electric heat load. The RES ratings is usually somewhat higher.

The coil voltage is rated for the voltage of the control circuit which is generally 24 VAC with most air conditioning systems.  I have added 2 red lines in the picture for the contactor on the left showing the 24 VAC connections for the contactor. As you can see the contacts are clearly separated from the main power contacts. The operation of the contactor is 24 VAC is applied to the coil via a common wire from the transformer. The other side of the coil is powered by the R side of the circuit through the thermostat. This energizes the coil and the magnetic field of the coil pulls in the contacts so they touch to start power flowing for the load. Power wires will be connected to the 2 terminals on top of the contactor and the compressor wires will be connected to the 2 terminals on the bottom of the contactor. When the contacts touch the compressor should start. Note the power wires and compressor wires can be installed on the top or bottom of the contactor.  As long as they are wired properly the system should operate.