An evaporative cooler (also swamp cooler, desert cooler, and wet air cooler) is a device that cools air through the evaporation of water. It is a very simple system and is popular in dryer climates since it also adds humidity to the air improving personal comfort.
Courtesy of Wikipedia
The typical swamp cooler is a metal box with removable louvered panels on all 4 sides. Installed in the vented side panels are cooling pads.  These pads can be made from a variety of materials.  In the bottom of the metal box is a water supply and water pump.  Water fills the bottom of the cooler and the pump pushes water through tubes that direct water to flow through the pads on the sides.  A blower motor in the center of the cooling unit pulls hot dry air from the outside, through the moistened pads and into the interior of the house through a vent in a ceiling or wall. Heat in the air evaporates water from the pads which are constantly re-dampened to continue the cooling process. 
Unit opened up

 Check the condition of the cooling pads.  Your cooler will run more efficiently if the pads are clean.  Consider replacing the cooler pads every 6 months, if you’re only using your cooler during the summer months this might mean you only need to change them every other summer.

Pump and float


Check to be sure the motor, pump and float are working properly.  If the float sticks the water will not shut off causing the unit to overflow, wasting water and potentially causing damage to adjacent structures.

 

Water supply and valve

Check the water line and all connections to be sure there are no leaks.  Especially if your unit is located on the roof and the water line runs through your attic space.
Check the water level.  The water should be about an inch below the top of the bottom tray.  If the water level is too high or too low adjust the float.

    Remember to clean out the entire unit and oil any moving parts prior to shutting down or starting up the system each season.