A hot, greasy pan should not be put straight into cold water because water hitting the hot surface instantly turns to steam and splashes the hot oil onto your hands and face. Sudden cooling can also damage the pan: thin pans warp, cast iron can crack, and enamel or nonstick coatings can be damaged. Take the pan off the heat, let it cool until warm, remove the oil, and then wash it.
The first hazard is splattering. After frying or sautéing, a pan can still be hundreds of °C / hundreds of °F hot, with hot oil at the bottom. When the pan is held under a faucet or placed in a sink full of water, the water gets underneath the oil, boils instantly, and expands into steam. The steam sends oil upward and sideways; it most often splashes onto the hand holding the pan, the arm, and the face. The dense burst of steam can also cause burns on its own. The sharp sizzle when a hot pan is put under the faucet is the sound of water turning into steam very quickly.
The second kind of damage affects the pan and is called thermal shock. Hot metal is expanded; cold water rapidly cools one side while the other stays hot, causing the metal to contract unevenly. In thin aluminum and steel pans, this can make the base bulge or warp; a warped pan no longer sits flat on the stove and will not heat evenly again. Cast iron can crack when cooled suddenly. Enamel can crack and flake off, while repeated thermal shocks can shorten the life of a nonstick coating.
The right order is simple. Remove the pan from the stove, set it on a heat-resistant surface or a cool burner, and let it rest until warm; this usually takes 10 to 15 minutes. Once the pan is cool enough to handle, wipe out the oil with a paper towel or pour it into a lidded container. Do not pour oil down the sink; it solidifies in the pipe as it cools and can cause a clog. Wash the pan with warm water and dish soap. Once the pan is warm, the handle will have cooled too and can be held safely.
Turning the browned bits at the bottom of the pan into a sauce is different and should not be confused with cooling a pan in water. After searing meat, drain off the excess oil, add a few spoonfuls of broth or water while the stove is on medium heat, and scrape up the browned bits with a spoon. There is little liquid here, most of the oil has been removed, and the pan is on the stove at a controlled temperature. Steam will still rise at first; pour the liquid into the pan away from yourself, and do not hold your face over it.
Sudden cooling is not needed for stuck-on residue either. Once the pan has cooled to warm, add warm water and a little dish soap and leave it for half an hour; the residue will soften on its own. If it is a cast-iron pan, clean it while warm with hot water and a brush, dry it right away, then heat it on the stove for a few minutes to finish drying; cast iron left soaking in water will rust. The same rule applies to glass baking dishes: do not set a hot dish straight from the oven on a wet or cold counter, because sudden cooling can crack the glass. For the same reason, do not pour a large amount of cold liquid into a very hot pan all at once.