When you are removing urine deposits whether it is fresh or old you are also removing the urine odor with it. The problem is in the case of a carpet and pad or a mattress or upholstered furniture etc. it becomes almost impossible to remove the entire urine residue. On hard surfaces such as sealed hardwood, tile or vinyl where the urine does not penetrate the surface, it is much easier to remove the majority if not all of the residue.
So step one is to remove as much urine as you can. Whether you use extraction with a wet dry vacuum or absorption with paper towel get as much out as possible. If you have access to a carpet or upholstery machine it will spray a solution onto the area and extract it back out again flushing the area and removing more urine odor. You can do this with absorption also by taking a spray bottle and spraying a little water or solution and absorbing it out again. If you do this be careful not to use too much liquid and drive the urine residue deeper into the carpet or fabric.
After you have removed what you can then you will treat the remaining residue with a good urine odor neutralizing chemical. You can read about the types of chemicals that are affective on urine odor removal here.