You will need to replace your toilet wax ring in two situations. The first is when you are swapping out the old toilet for a new one in which case the old wax ring will have to be removed to install a new one. The second is when there is a leak at the bottom of your toilet.
Step 1 - Turn the Water Off
Since this operation requires you to lift the toilet off of its seating, you will have to disconnect some water pipes. The first operation then, is to shut off the water with the water valve in the rear of the toilet near the wall. Flush the toilet a few times to get rid of the water in the tank and bowl. Remove the supply line that goes from the valve to the tank. Use the adjustable wrench to loosen the compression nut. Be careful not to bend it. If you do it will not fit right.
Step 2 - Remove the Toilet
On each side of the toilet tank is a bolt that holds the toilet to the floor. The bolts will have coverings of some kind. Flip the covers over to expose the mounting bolts. Loosen the nuts with an adjustable wrench. Once the nuts are removed from the toilet, lift the toilet straight up off the bolts. Since a toilet is much heavier than it looks and has an awkward weight distribution, you will probably want to have someone help you with this part.
Step 3 - Remove Old Wax Ring
You will immediately notice the wax ring on top of the drain pipes and flange. Scrape the old wax ring off of the drain. Use a flat scraper and be careful not to damage anything else while taking the ring off. If the ring has been in place for several years, it may take a little bit of force to remove. Remove the old bolting hardware.
Step 4 - Install New Ring
Place the new wax ring to the area and gently press it into place. Do not cause any misshaping, but make sure it is seated in the correct position. Install the new bolts.
Step 5 - Reinstall Toilet Seat
Lift the toilet up and place it straight down over the bolts and wax ring. Do not slide it on. Lift the toilet straight up and gently place over the ring. Apply even pressure around the entire ring and press the toilet into it for firm contact with the floor. Tighten each bolt slightly, moving around the toilet until the toilet does not wiggle or move. Do not tighten one bolt all the way and then tighten the next. Do it in steps, tightening the bolts evenly until the toilet is snug. If you overtighten the nuts on the bolt, you will compress the ring, causing it to split or warp.
Step 6 - Connect Water
After the toilet is correctly situated back on its mounting hardware, you can connect the water supply. Use the adjustable wrench and do not overtighten it. Turn the water back on and let the bowl fill up. Flush and check it for any leaks.