How to Patch a Flat Tire on a Bicycle

Judith Guthoerl demonstrates how.
First she loosens the nuts holding the flat front wheel on.
You can actually fix a flat without removing the wheel, but it usually isn't easier that way.

The wheel hangs up on the brake pads.
The brake calipers are too close together to get the wheel out.
She pulls the end of the brake cable out of the opposing bracket thingy so the calipers can pop apart, letting the wheel out.
Every type of brakes has a different way of doing this.

The bicycle patch kit came with these three little levers. They're called "tire irons" for historical reasons. She uses one to pry the tire over the side of the rim and then hangs the little hook on a spoke to keep it in place. She slides the next iron under the tire and pries it out further. She repeats with the third iron.
Now there's enough tire over the rim that it's easy to pull the rest over by hand.

Spoon handles work if they're smooth and you don't have official tire irons handy.
If you use screwdrivers it's very easy to puncture the innertube.

Remove the innertube from the tire.
You don't have to pull the tire all the way off the rim to do this.

Whatever punctured your innertube might still be impaled in your tire. If you don't find and remove it, you'll get another flat right away. Look and feel inside the tire to see if there's something poking through. It's usually a nail, thorn, or shard of glass. Don't cut yourself.

Put some air in the tube, but don't overdo it.
This air chuck is nice because it has a built-in pressure guage.

Put some water in the sink, or a bucket. A river or lake would be fine also.
If your sink has no plug stuff a plastic bag in the drain.
Submerge the inflated tube. Look, listen, and feel for air escaping.
Don't stop checking the tube after finding one leak. There may be others. If you find a few holes near each other, there's probably a thorn or nail still embedded in the tire.
If there's a small slit near the rim, you probably just did that by prying too hard with a screwdriver while getting the tire off.
Check the valve stem. They get leaks also.
Dry off the tube and draw a circle around the leak.
Make the circle a bit larger than the patch you'll use. Now you won't lose the leak. I've patched tubes many times to find that the patch wasn't over the hole.

The patch kit comes with a piece of sandpaper or a tiny cheesegrater looking thing.
Use it to roughen an area around the leak that's a bit bigger than the patch you'll use.

Is what the patch cement is called. Regular rubber cement or contact cement won't work as well. They aren't nearly toxic and explosive enough. Your friends will steal it from you and you'll never know the real reason.
The new tube of fluid is sealed. There's a spike in the top of the cap. Use it to puncture the seal.

Smear a thin layer of vulcanizing fluid all around the leak, covering the area where the patch will go.

Set the tube aside and let the cement dry for a few minutes.
Read the directions that came with the patch kit if you haven't already.

It's the hotter than any day last year and it's only May.

Peel half the backing off the patch. There's a lot of variety in patches. Some of them come with slightly different directions. It probably doesn't matter.
Press the patch down over the leak so that you don't trap any air bubbles. Peel the rest of the backing off as you do this.
Use the handle of your Rambo knife to mash down the patch and press it fully onto the tire. If there's a cellophane backing on the patch, scrape it off.

Judith's last name means "good listener" in German.
She's already showed you the skills to do the rest of these steps:

1. Inflate the tube, bubble test it for leaks to see if you really fixed it. Dry it off.
2. Deflate it slightly but still leave some air inside so you won't pinch it with the tire irons.
3. Put it in the tire and shove the valve stem through its hole in the rim.
4. Push the tire over the rim with your hands. If necessary use the tire irons to finish prying the tire onto the rim.
5. Inflate the tire. The proper pressure is printed on the side of the tire. To be extra sure you've fixed the leak without making new ones, you can bubbletest it for leaks when it's on the rim.
6. Put the wheel back on the bike. Those weird washers hook into holes near the end of the fork.
7. Reconnect the brake cable.

Congratulations! You've fixed a flat!
You're now back on your bike, which makes you the more efficient than any animal on earth.
If it weren't for your wastefully big brain you'd hardly need any food at all!