What's the best technique for removing a tyre from the rim?
First off, you usually only need to remove one side of the tire if you're just fixing a puncture. 


If the tire isn't fully deflated, let all the rest of the air out. It'll make your life easier. 
Using your hands, grip the tire near the valve and wiggle it a bit to loosen the bead up from the rim. Work your way around the tire, repeating this process until you get back to the valve. If you've done this and step 1 properly, the tire should be loosely flopping around inside of the rim. 
Grab three tire levers. You can do the job with less, but it's harder. I've never needed four. The ones that will get tire off most easily will have a hook one one end, like these: 

Insert one tire lever underneath the bead just above the spoke that is most directly opposite of the valve. The tire will be a little looser (more loose?) directly across from the valve and thus, easier to remove. Pry it up and hook the other end of the lever on a spoke. 
Check to make sure that you've only got one side of the bead under the lever. If you've got both sides of the bead, the rest will be somewhere between very difficult and impossible.
Repeat step four on the spoke to the right of the one you just hooked, and then the spoke on the left. 
At this point, the tire bead may be loose enough that some of the tire levers fall off. If so, great. Pry it the rest of the way off with your finger. If none of the levers has fallen out, it's ok. You just have a tire that fits a bit more tightly. You should still be able to get your finger in just to the outside of your three spokes and slide it along to pry the tire off. As I mentioned before, I have never had to use a fourth lever.
By now, you should have the bead completely removed on one side. You can remove the tube by reaching into the tire opposite of the valve and just pulling it out. You'll reach a point where the valve is the only thing holding it in place. Push the tire out of the way and remove the valve. 
If you''d like to completely remove the tire, you'll be able to just yank it off the rest of the way with a firm grip.


The steps above will work on any tire. Skinny, hard case, non-foldable tires tend to be the hardest to get off. Fatties that are foldable can often be removed with nothing more than a good yank. If you have the option, it's still better to do it with tire levers though so you don't risk pulling too hard and damaging the tube. Tires inbetween those ranges tend to vary in difficulty.