Luckily, there are 2 things that help tires get onto and off of rims;

1. The tire is not perfectly rigid.

2. The rim has a special area called a 'dropped center' that one side of the tire can be slid into, which lets the tire go eccentric to the rim. This combined with the give of the tire allows it to slip out.

Now here's how to get it to work:

1. Unscrew the valve from the stem and let the tire go completely flat. Leave the valve out (but don't lose it if you are going to reuse the tube).

2. Lay the wheel down with the valve stem at the 12 o'clock position.

3. Push down on the tire adjacent to the rim at the 6 o'clock position to pop it loose from the rim. Do this all the way around the wheel so it's loosened up.

4. Press down on the tire at the 6 o'clock position (I kneel on it); make sure the rim of the tire is close to the center of the rim where the spokes attach - when you start prying, the tire needs this room to slide into. While holding the tire with your knees or whatever, insert one of the tire irons flat ends just under the tire's edge at the noon position - I.E. you're kneeling on the 6 o'clock part, and prying the 12 o'clock part over the rim to the outside.

5. Once you get a small amount pried over, use the other tire iron and do the same thing a couple of inches to the left or right of where your first iron is holding it's place - in a bit you can worry that part of the tire over the rim.

6. Keep moving the tire irons further apart and pry a bit more of the tire over. Once you get past halfway it speeds up a lot. Soon you will have one edge of the tire past the rim.

7. Reach inside the tire and pull the tube out; you'll have to wiggle it a bit to get the valve stem out.

8. Using the same approach on the other half of the tire, 'walk' the lip over the rim. Usually, once it reaches halfway you can just pull the wheel out the rest of the way since you are no longer fighting both sides of the tire.

9. Inspect the wheel, make sure the rubber liner is in good condition and all spokes are covered (or you'll be doing this again very soon!). Duct tape CAN be used for a while, but I think a real rubber liner is best.

10. Mounting the tire is pretty much the reverse procedure. Put the 6 o'clock part of the lip over first so it can get to the dropped center of the rim. It's hard to do this in the valve stem area, which is why I laid it out as I did.

Some notes and cautions:

Have an extra inner tube on hand. They don't cost much, and most people pinch one doing this at one time or other.

It IS possible to insert the valve stem by hand; however, there is a tool that has a metal cap on a cable to pull it through. It's worth the couple of $$!

When you insert the tube after getting the first lip of the tire on, put a little air in the tube first. Don't blow it up big, just enough air to make it easy to handle and pull the edges away from the rim.

Soapy water helps the tire slip on and off. Some have mentioned liquid KY jelly, I've never used that.

Warren Johns has advised putting the first tire rim on past the 2nd rim of the tire so that you have the wheel essentially sitting inside both rims of the tire - this makes it easier for him to handle the tube and reduces chance of pinching the tube. I like the idea but my hands are too big to make this work for me.