Whatever the source of the urine stains -- from a beloved family pet or your favorite toddler -- the trick to cleaning them out of leather is to act fast. The sooner you treat the stain, the better your odds of removing. Cat urine is the most difficult odor to remove, so methods that work for cat urine also work for human urine and stains from dogs and most other pets.
If you find the stain while it's still wet, use a white paper towel or cloth to absorb any urine still puddled on the surface of the cushion. Be sure to blot the stain and not rub it, as rubbing will spread the urine around and make the problem worse.
If your couch allows, unzip the cover over the cushion and remove the padding inside. If any urine has reached the stuffing, take it to a large sink or your bathtub and wash it thoroughly with an enzymatic cleaner, the type that is readily available at pet stores. Squeeze and smoosh the stuffing to work the cleaner in and then repeat the process with plain water to rinse the cleaner away. Squeeze the stuffing to drain as much water as you can and press towels into it to absorb as much water as possible. Set the stuffing outside in the sun to dry if possible or place it in a well-ventilated room with a fan blowing across it.
Wipe down the leather cushion cover with a white damp cloth that you've wet with enzymatic or homemade odor-removing cleanser to remove the smell and urine stain. Do not saturate the leather; instead, use as little liquid as possible. Be sure to wipe the entire cushion from seam to seam so that you don't create a discolored spot when the leather dries.
To keep your leather supple, allow the leather cushion cover to air dry completely after washing and then treat it with a dab of your favorite leather conditioner. This will restore any moisture your stain removal solution may have stripped from the material.
Once it has thoroughly dried, place your couch or chair cushion stuffing back into its leather cover and place the cushion back on the couch.