White vinegar and ice water. Apply white vinegar directly with a clean cloth/sponge. Let set about 10 minutes. Then with a clean damp cloth/sponge dipped in ice water, dampen the area. Use a dry cloth or towel to soak up the liquid. Repeat the ice water if the vinegar smells remains (it shouldn't). Make sure you get it as dry as possible. The vinegar does something to the urine (breaks up the proteins, I think) so that the smell is eradicated. The ice was gets rid of the vinegar smell. A dog breeder taught this to me over 20 years ago. Works on cat urine, too.

Baking soda and vinegar. Sprinkle liberally with baking soda then soak with vinegar. Let sit over night.
Scrub with water, let dry. Vacum.

It depends on how fresh is the stain, and is there stain at all or you can just smell the odor.
If you are working with a fresh urine puddle, blot the spot with paper towel and spray it with mixed vinegar and warm water. This should “kill” the smell. You can mix equal parts of water and white vinegar.
In case the urine left a stain you can put some baking soda and water on the spot. Once it dries just vacuum the powder from the spot.
For even stronger odors and stains you can check
http://www.cool-small-pets.com/homemade-pet-odor-r emoval.html

The solution I've used with success is a combination of the above with an addition. We used this in an elderly care home I worked at when one resident kept having urinary accidents on the rug and I've done it in my home when my pets had accidents.

Get the spot dry using paper towels or rags.
3 parts water - 1 part vinegar -- pour over soiled area and let sit 5 - 10 minutes.
Get the spot dry using paper towels or old rags.
Sprinkle with baking soda.
Mix hydrogen peroxide with dish liquid: 3/4 cup of peroxide to 1 teaspoon detergent.
Soak the area with the liquid and wait till it dries completely, then vacuum.