Some of the tools pictured above will be required to replace a tire.  Since most tire replacements are a home maintenance project, in addition to the above tools, it is handy to have a robust tire pump handy as well.  

Tire replacement happens because of a damaged tire, a worn out tire or the replacement of a tire for increased riding performance. Rarely would the tube be replaced or need to be replaced when doing a tire replacement. Tubes are repairable and do that before replacing a tube.

The first thing you need to do when replacing a tire is to remove the wheel from the bike.

Grabbing the derailleur.

Pulling it back to clear the rear sprockets.

Once enough air has been removed from the tube, insert your first tire removal lever as shown in the above photo on the left and hook it on a spoke as shown.  The next tire lever should be inserted about 4 to 6 inches away from the first and again the tire should be levered over the rim.  Repeat until one entire side of the tire is free of the rim.  Note that it does not matter where on the tire rim you start this tire removal lever process nor does it matter which side.  The only object is to remove the tire from one side only, completely, so that the tube can be removed.

Once a complete side of the tire is free of the rim, then starting with the tire valve, remove the tube from the wheel.  If it is a Presta valve, first remove the nut holding the Presta valve to the rim as well.  Once the tube is removed entirely, then wrestle the old tire the rest of the way off of the tire rim.

Now grab the new tire.  Almost every tire manufactured for bikes, has an arrow on it indicating the direction of rotation.  Locate this arrow and orientate your tire such that when the tire rim spins on your bike forward against the road, the tire rotation arrow on the tire will match and be pointed in the same rotation direction as the rim.  If it helps you, mark an arrow on the rim in the direction of rotation and see if it matches your tire rotation arrow.

Can't find the arrow?  In recreational riding, I just don't think this directional rotation arrow proves a whole lot. That being said, if you ride with knowledgeable bicyclists and they know you replaced the tire, they will probably look to see if the arrow is going the right direction.  If they find it isn't, they will tell you to take the tire off and rotate it 180 degrees to put the arrow facing the correct direction.  If you ask them why, they won't be able to tell you.  Just pretend to hear their sage advice and promise to make it right the next time you get a chance.

You will also notice that this specific tire says "front use only."  Yes, higher quality tires do differentiate between the front and back with their tires, optimizing the tread and tire casing for the task at hand.  For instance, the front tire receives almost all of the braking abuse while the rear tire takes a lot of sideways abuse.  Tire abuse can be compensated for with optimised tire construction.  I believe as long as you don't enter a race at Moab, Utah, you will be fine with any installation of any tire.  If you are an aspiring racer or serious mountain biker, then pay attention and put the directional arrow going the correct direction and a "front" tire on the front and a "rear tire" on the rear.  Also, replace the front and rear tires with the same brand and model tire because those front and rear markings and directional arrows benefit you the most when the front and rear tires are matched.

Now that I have given you all of the whys, lets mount your new tire.  With the directional arrow positioned correctly, ease any edge of the tire over the rim leaving the other edge free to place the tube into.  

Install the tube valve first and if it is a Presta valve apply the nut to the stem as well.  If it is a tube with a Schrader valve, make sure you reinstall the valve with the valve tool in the opposite direction you took it out. Make the Sharder valve finger tight, being careful not to over tighten it.  

With the tube valve installed in the rim, now work the tube completely into the tire making sure there are no folds in the tube. It helps to have the tiniest bit of pressure in the tube while doing this.  A tube that is completely flat will not conform to the tire properly when you lever the tire back over the rim.

After you have stuffed and poked and herded the tube back inside the tire, pick a point on the tire and start placing the remaining tire edge back onto the inside of the rim.  You can usually do this by hand except for the last six or nine inches.  If you can, force the tire over the outside of the rim with your thumbs.  If that is not possible then carefully use a tire lever to lever the last few inches over the outside rim and into the inside rim.  If you have to use a tire lever, be aware that it is very easy for your lever to catch an edge of the tube, pinching it against the rim and causing a leak. You wont know that you caused this leak until after you try to pump up the tire and you hear the ominous hissing of air escaping from the leak that you just caused.

After the tire is seated, inspect it to make sure the tire bead is below the rim at all points on both sides of the wheel. It is not unusual for the tire bead to be above the rim if something is blocking it; for instance.  The tire valve is the most common reason for this.  If the tire valve is causing the tire rim not to seat, physically push the valve up into the tire until the tire bead seats and then pull the valve back down.  If the tire bead is not seated in any one place on either side of the rim when you pump the tire up, the tube will sneak out and literally blow up. It may do this right away or after you have jumped on your bike and have gone down the road.

Pump up the tire and reinstall the wheel on your bike reversing the take off procedure.  Remember that the derailleur will be the first thing in the way and you will have to physically pull it back in order to clear the sprockets. Retighten the lever.  First tighten the thumb screw opposite of the lever until you feel the lever working when you apply it.  Be sure it is reasonably difficult to push the lever closed as this is what holds the wheel to your bike.  If the lever is loosely applied, your wheel will become loose while riding.

If you have rim brakes, reinstall the brake cable and test the brakes before using them.