This article goes along with our other articles about fixing flat tires.
Inner tubes: choices, valve types, tips and tricks
Tools and supplies for tire maintenance
A flat tire may be repaired either by replacing the inner tube or by patching it.
To replace the inner tube, you must remove the wheel from the bicycle. But to patch the inner tube, you need only expose it, without removing the wheel.
A wheel with axle nuts is harder to remove and replace than one with a quick release. Even harder is a wheel with an internal-gear hub or hub brake; worse, a wheel with both; yet worse, a bicycle with a full chain case; and worst of all is if you don't have a wrench to turn the axle nuts. When fixing a flat out on the road, you do whatever works best. Also, you may be called on to fix a flat for someone else.
So, to keep all options open, I carry a tool kit including a patch kit, multi-tool and adjustable wrench, even if I'm also carrying an inner tube. The patch kit also comes in handy if I flat more than once.
The videos embedded in this article show how to remove and replace the inner tube for patching without removing the wheel.
Here's the video about removing the inner tube. The only tools you need are tire levers, or maybe not even tire levers if the tire is a loose fit. You will, of course, need a patch kit to repair the inner tube. Check our article on flat tires for advice on inspection for the cause of the problem, and on patching.
And here's the video about replacing the inner tube. This video also gives some tips about pumping technique.