Sewing on a button is very simple once you know how to do it! I'll teach you how to sew on both shank and flat buttons so you're covered anytime one falls off. :)

Step 1: Types of Buttons.

Flat buttons are the most common kind - they're what you'll find on dress shirts and most clothing. They have two or four holes, depending on the style.

Shank buttons have no holes on top, but they have a small protrusion on the back that is hollow at the end. They're more often found on dresses and coats.

Step 2: Things You Need to Sew on a Shank Button:

needle
8 inches of doubled over thread - knot the ends a few times!
shank button
pen/pencil to mark your sewing spot

Step 3: Sewing on a Shank Button, Part One.

bring your needle and thread up through the fabric on the spot you marked
pass them through the hole in the shank
bring them down through the fabric in the same spot

Step 4: Sewing on a Shank Button, Part Two.

pull the thread tight so the bottom of the shank sits against the fabric
continue passing the needle and thread through the fabric and shank, at least 8-10 times.
bring the needle to the back side of the fabric on the last pass.

Step 5: Sewing on a Shank Button, Part Three.

on the wrong side of the fabric, knot the thread several times, anchoring the knots with small stitches.
cut off your ends and enjoy your button!

Step 6: Things You'll Need to Sew on a Flat Button:

needle
8 inches of doubled over thread - ends knotted a few times!
button
toothpick
pen/pencil to mark your sewing spot

Step 7: Sewing on a Flat Button, Part One.

start by marking the spot where you'll be sewing.
bring your needle up through the fabric next to the spot
bring the needle and thread through one side of the button
then bring it down through the other side of the button and through the fabric next to the spot.
now you have it anchored!

Step 8: Sewing on a Flat Button, Part Two.

push a toothpick between the fabric and button, between the two holes
bring the needle up and down through the holes of the button and through the back of the fabric several times - I do this at least 8-10 times in most cases.
on the last time, bring the needle up through the fabric next to the stitches that are already made, but not through the button (picture 3)
wind the string counter clockwise around the thread going from the button to the fabric a few times, pulling it tight and leaving the toothpick in place.
remove the toothpick and wind it a few more times
it should look like the last photo when you're done!

If anyone is wondering why anyone would take the time to do this - it's so your button will not break off and your clothes will fit better. There needs to be a little space between the button and the fabric so that an additional layer of fabric can fit easily between them. Otherwise, buttoning will be a chore and the fabric will pull in odd directions. And your thread will wear very quickly with all the friction... meaning a button that will not stay on long.

Step 9: Sewing on a Flat Button, Part Three.

after winding, bring the needle to the wrong side of the fabric and knot well by making small stitches in the fabric and then knotting.
knot at least three times
knot security is awesome!
trim your ends and marvel at your fantastic sewing abilities.

Step 10: Additional Hints and Tips:

always, always double your thread if using regular cotton or polyester thread. If you're using something more heavy duty like embroidery floss or upholstery thread, it's okay to use a single strand.
if you'd like to further secure your knots, a dab of fabric glue (my favorite is Fabri-Tac) will do it.
make sure that all your stitches are hidden under the button - otherwise it'll look messy and they could get snagged and break.
if you're sewing on a flat button with four holes, anchor the top two holes first and then move to the bottom ones.

And if you need help with general sewing, please check out my instructable on how to sew. It'll cover most everything you need to know!