Remove the wheel containing the defeated tube. It may be advantageous to start by placing the bicycle within a home repair stand, held securely by the seatpost. Many types of brakes can be easily released to aid in tire repair, so be sure to disengage the brake, if necessary.
Most modern wheels remove by means of Quick-Release (QR) skewers, pictured here, which pass entirely through the hubs, and have open and closed positions to secure them to the frame.
To remove the wheel, flip the lever to the "open" position. Hold the non-drive side nut of the skewer as you turn the lever counter-clockwise to loosen. Oftentimes, the wheel can be removed without completely unthreading the skewer.
Note the orientation of the two springs and skewer (it enters from the non-drive side) for ease of reinstallation.
Many other wheels will use a bolt-on standard, usually a 15mm bolt on a threaded axle. Use a 15mm or adjustable wrench to remove these axle bolts by turning them counter-clockwise.
Some higher-end bicycles have wheels that connect by means of a thru-axle (TA) system. Like QR skewers, Thru-Axles run internally through the entire length of the hub. Thru Axles, however, thread into a special sleeve built into the fork or frame dropout. Removing these is as simple as turning the TA anti-clockwise, although each manufacturer may have special instructions.
Using the tire levers, unseat the bead of the tire by running the lever along the inside of the rim. With the first bead unseated, repeat the process to remove the second bead off of the rim. Examine the outside for the cause of failure.
The bead is the part of the tire designed to hold the tire to the rim. For more about this step, see how to replace a bicycle tube.
After looking around the outside, remove the tube from inside the tire. Run your fingers lightly along the inside, checking for any sharp spines or staples. BE CAREFUL! THESE CAN BE VERY SHARP AND CUT YOUR HAND.
If you are having difficulty finding the hole, lightly inflate it and check by submerging under water. Escaping air bubbles will point to the puncture. Once you have found the problem, use a metallic sharpie to mark the area.
Apply the tube cement and smear it evenly over the scuffed area using a clean finger, or foil edge of the patch. Make sure to spread enough to cover the entire affected area.
Choose an appropriately sized patch so as to completely cover the affected area. Remove the foil backing from the patch. Leaving on the transparent plastic covering, place the patch over the area, pressing the tube and patch together.
Holding firmly between your thumb and forefinger, apply moderate pressure while holding the patch in place. This should take about 5 minutes. Leave the transparent plastic film on as it will help to resist abrasion.
Pump up your tube a small amount, just enough to give it shape. Place the lightly-inflated tube within the tire. Using only your hands, roll the beads of the tire (one at a time) onto the rim until the tire is properly seated. Take care to install tires with the proper direction of rotation, when applicable. An arrow indicating direction of rotation can be found on the bike tire itself.
Once seated, pump up the tube slowly to the desired pressure, checking for irregularities in the tire. If either bead is not seating properly within the rim, deflate the tube, lubricate the beads of the tire with some soapy water, and reattempt inflation.
Using your wheels' chosen standard of axle binding, tighten by turning clockwise.
If using a QR skewer, hold the drive side nut while turning the lever clockwise. Close the lever from the open to closed position, ensuring that the clamping action secures the wheel within the frame.
After the bicycle has been sitting for 15 minutes or so, use the air pump to check the inflation pressure of the tire. If it is the same, test ride the bicycle a short distance and then re-check the pressure. If it shows no signs of loss, it's good to go!