Tubeless mountain bike tires are not just a marketing gimmick-- they have many benefits. For large volume tires, such as those used on mountain bikes, a tubeless conversion will allow running lower tire pressures as you no longer have to worry about "pinch flats". Lower tire pressures yield better traction but will increase rolling resistance. Additional benefits of tubeless tires include puncture protection (the sealant will continuously seal small punctures from thorns), and some weight savings for the weight weenies out there.

Things you will need:

A pair of MTB tires (it helps if they are tubeless ready but not necessary)
Gorilla tape ($6, home depot)
Tubeless valve stems with removable core ($12.50, ebay)
Tire Sealant ($14, ebay)
Exacto knife or razor
Pump or air compressor
Needle-nose pliers
Tire levers

Let's start...

Step 1: Remove Tires and Tubes

Release the air pressure from the tires.
Attempt to remove the tire by hand. Often times, this is not possible and tire levers must be used.
Carefully pry out one side of the tire over the rim using tire levers. Make sure the tire lever is not pinching the deflated tube as we will be using the tubes later (it also helps to have spare tubes around).
Once you have freed one side of the tire, remove the existing tube and carefully pry out the tire.

Step 2: Remove Existing Rim Tape

Most wheel sets come with rim tape to protect the tube from sharp edges and spoke ends. Since we are removing the tubes, we no longer need the rim tape. Pick at the edge of the tape until it is loose. Feel free to cut the rim tape if it is giving you trouble. Thoroughly scrub the inside of the rim with soap and water and allow for it to dry.

Step 3: Applying Gorilla Tape

Measure the outer width of your rim.
Using a razor or knife, make a small cut 2mm less than the rim's outer width on the gorilla tape roll. The goal here is to start a strip of tape that is slightly larger than the internal width of the rim and just barely overlaps the bottom portion of the tire's bead once inserted.
On the inside of the rim, locate the hole where the valve stem used to be.
Peel back about 2" of the Gorilla tape and start applying the tape 2" before the valve stem location. We will be finishing the circle 2" after the valve stem so that there's a 2" overlap on each side of the valve stem.
Start applying the Gorilla tape by peeling 4-5" from the roll and ensuring each section is centered. Do not make any cuts to the tape. The first cut should be enough to initiate the desired width of the tape and you should be able to tear straight along that line. You want to have only one, clean continues strip of tape.
Run your thumb through the groove with each pass to make sure that the tape makes contact and sits flush with the rim.
Continue going around the rim with the Gorilla tape. Adjust how much tape you are peeling from the roll as you get more comfortable.
Make your final cut to the tape once you overlap your starting point by 4".
Run your thumb all along the groove one more time to ensure the tape is seated properly.

Step 4: Inserting Valve Stems

Often times, it is difficult to pump up non-tubeless tires without a tube for the first time. Inserting the tube and mounting the tire again can help get the tire bead seated. When removing the tube the second time, be extra careful to only remove one side of the tire bead, leaving the other bead seated in place. This helps pumping the tire once you go tubeless. To do this wee need to:

Locate the original valve stem location on top of the Gorilla tape. Remember, this should be centered in the area where the Gorilla tape overlaps.
Using a razor or Xacto knife, make two small incisions in the shape of an "X" so that the valve stem can be inserted into the hole.
Insert the valve stem of the tube into the hole and thread the nut to secure the valve stem in place.
Mount one side of the tire over the tube and to the desired position (pay attention to tire direction as this will be the final time we mount the tire)
Carefully feed the tube in the tire, making sure there are no twists.
Mount the other side of the tire and ensure that the tire bead does not pinch the tube. It may be necessary to use tire levers again.
Pump up the tire to recommended pressure.
Deflate the tire making sure the bead remains seated on the rim. It helps to have the wheel resting sideways so that no weight is resting on the tire once deflated.
Carefully deflate the tire and dismount one side ONLY.
Remove the tube from the dismounted side and insert the tubeless valve stem.
Mount the tire one final time.

Step 5: Filling With Sealant and Pumping

Attempt to pump up the tire without any sealant. The first time, you may have to pump rapidly or use an air compressor. Basically, you need to get both beads seated on the rim in order to create a temporary seal. This will make pumping much easier. Some tires will be more difficult to pump than others.
Once you have successfully seated both beads, remove the valve core from the stem using needle-nose pliers. The valve core has two flat section in opposing sides along the threads which you can use for grip. Try to not rest the wheel on the tire from this point on. It is best to work with the wheel resting horizontally.
Fill the syringe with 2 oz. of sealant.
Insert the syringe filled with sealant over the valve stem and squeeze until all of the sealant is transferred from the syringe to the tire.
Shake the tire in all direction to ensure that the sealant has thoroughly coated the insides of the tire.
Pump the tire to manufacturer recommended pressure while it rests horizontally with no weight resting on the tire.
Shake the wheel in all directions to ensure the sealant has coated the insides once more.
Check the tire pressure the following day. The tire might have lost some pressure while the sealant was drying. Pump to desired air pressure-- I find 20-25psi to be the sweet spot.
Typically, sealant should be reapplied every 6 months (more frequently if you live in hot and dry climates).

Happy riding!