If the urine on the carpet is still wet, use paper towels and newspaper to soak up as much as possible. If it is dry urine, use a wet vac or extractor to pull all traces of the stain out of the carpet.
Stack a thick layer of newspaper underneath the urine spot if it is still wet and it's possible to pull up the carpet. If not, begin with a thick layer of paper towels on top of the wet spot. Put a thick layer of newspaper on top of that, and walk on the newspaper, pressing down into the rug. Pick up the layers when the dampness spreads through them, repeating until the carpet is almost dry.
Use a wet vac to clean away signs of heavy carpet stains. Rent one at a local home improvement store if you do not have one or want to buy one. Follow the instructions in the owner's manual, and vacuum until the stain appears to be gone.
Spread a pet odor neutralizer after you've cleaned the rug, either using paper towel and newspaper. Stay away from steam cleaners, as the heat sets the odor and stain into the carpet. Cleansers with strong odors, such as vinegar and ammonia, should be avoided, as they sometimes encourage pets to fight that odor by adding more of their own urine.