Step one: If the urine you are removing is fresh and still moist the first step in removing it is to get as much out as possible before it dries. Two ways to do this is blotting / absorption or extraction / wet vacuum.
When blotting take a clean dry absorbent towel or rag and place it on the moist urine spot. Paper towel works well. Put gentle pressure on the towel being careful not to push the urine deeper into the fiber/fabric or spread it out any further. The purpose of this is to absorb the urine into your towel and out of your carpet, or whatever you are getting it out of. Turn your towel over or change the towel if necessary to keep a clean dry area absorbing as much urine as possible. When you have removed the majority of the urine, again using a clean dry area of towel place it on the spot and put a book (like a phone book) on top of it and let it sit. You want a book that covers the size of the spot and is heavy enough to apply pressure so the towel will absorb the last bit of moisture that it can. You can place a piece of plastic between the rag and the book so the book does not get moisture on it.
When removing urine using the extraction method use a wet dry vacuum or a home carpet cleaning machine. Use it to extract (vacuum) the urine out. Use the strongest wet vacuum available to you. Use it to pull the urine out of the fabric/fiber or off the floor. Extract as much as you can working from the outside edges of the spot to the middle. If you have a see through (clear) wand or tool on your vacuum you can see when the urine stops coming out.
If the urine is not fresh but has dried then skip this first step of extraction/absorption
Move onto the next step in removing urine of pretreat and clean.