Cleaning up urine properly is both unpleasant and important. Knowing the correct cleaners to use helps, but there are some understandable concerns about using bleach because of its tendency to react with the ammonia in urine. The simple answer is that most websites recommend using bleach as a cleaner, but take into account a few things.
Bleach is an incredibly powerful cleaner, capable even of removing color from clothing. Ammonia is also a common cleaning agent, it also happens to occur naturally in urine. Under a black light, ammonia actually fluoresces, which is helpful if you're trying to locate a hidden stain which is making the room smell. However, when chlorine from bleach mixes with ammonia, it can create a noxious gas used in World War I during trench battles: mustard gas. In small spaces, this gas can even be fatal, but small batches can make people lightheaded. To avoid making this gas while cleaning,concentrate on what you're doing.
Most sites recommend a 1 part bleach to 10 parts water solution in a spray bottle. That 1:10 ratio makes a cleaning solution that is both effective and safe to use. In that concentration, there isn't enough of the chlorine in the solution to react with the ammonia. Instead, it simply cleans and disinfects the area. Mop up the urine with a paper towel first, then apply the solution liberally on the area.
If pets or animals will be using the area soon, be sure to wipe the area a second time with your standard pet-safe cleaner. Before using this cleaning mixture on fabrics, it is a good idea to spot treat the area to see if the color will run. Also note that this solution will only work for 24 hours, and after that it loses its potency.
Whenever cleaning up urine stains, observe basic safety precautions to prevent the spread of illness. Wear rubber gloves while cleaning, and wash your hands thoroughly afterward. Be careful when cleaning in enclosed spaces, as the fumes from cleaning products can be unhealthy.
There are many non-bleach cleaners available if you are still not comfortable with bleach. Bac-Out and Nature's Miracle are two such cleaners; both use enzymes to eat odors and kill bacteria and are therefore safe for animals and the environment. Febreeze is also a popular and useful deodorizing product to eliminate urine odors.