How to Machine Bind a Quilt
There are a few tried-and-true methods for finishing a quilt. Traditionalists prefer hand binding, where the binding is machine stitched to the front of the quilt then folded over and sewn down to the backing by hand. Hand quilters will use a fun big stitch (it looks so sweet). But for those of us who can't do a lot of handwork or who are too impatient, there's a third way (and its variations).
Machine binding is a zillion times faster. (I timed it, I swear.) 
Some prep: if you haven't squared your quilt yet, do so. There will be a separate tutorial on that. Also, as with quilting, if you have a walking foot for your machine, now's the time to use it.
Determine How Much Fabric You Need
You'll need one loooooong strip that will fit around your entire quilt. If you know how to prep that, skip to "Attach the Binding".
Most patterns will tell you how much fabric you'll need to make this. It's usually based on using a 2.5" wide strip, but you can vary that depending on how thick or thin you want your binding to be. 
1. Add the length in inches of each side of your quilt together. Divide by 40, or the width of your fabric (likely/possibly 42" WOF), to figure out how many strips you need just for the perimeter. Round this up to the nearest whole number. 
If your quilt is 80" square, the math would be: 
80 + 80 + 80 + 80 = 320 / 40 = 8
Add 1 to this number to give yourself an extra strip for seam allowances and tails to join ends. So we'll need 9 strips for our 80" square quilt. 
2. Next we'll multiply how many strips we need by how wide the strips will be to determine how much yardage we need. Because fabric needs to be trimmed so it's straight (also known as trued), we'll add ~2-4" to this total to make sure we have enough fabric once we've trued it.
9 x 2.5 = 22.5 + (~3.5) = 26
So for our 80" square quilt, we'll need ~26" of fabric. 
26/36 = 72.2% (or rounded to 3/4 yard) 
I always err on the side of being generous; you can do things with scraps if you have too much, but no one wants to have to run back to the fabric store if they're short.
3. Cut your 9 strips the way you would any strips. Trim selvedges. We're going to join them on the bias so that they lay flatter and the seam is distributed across the binding. This is why we added an extra strip - to give us enough extra fabric for a 2.5" seam. 
Make the Binding 
4. Place two strips right sides together at a 90* angle to each other allowing a little bit of overlap for each piece. 
5. Sew from one corner to the other as shown in the diagram below, starting and stopping where the overlaps form corners. Cut 1/4" away from the stitch line to remove excess fabric as shown in the diagram below. Finger press the seam open and trim the dog ears. Repeat for each strip, adding it to the previous one, until you have one continuous strip of binding made from all the strips.     
(a) 
         (b) 
6. Fold the strip in half lengthwise and press along the entire thing. Yes, it's annoying and tedious. I'm sorry.
Attach the Binding
7. You're going to use a slightly bigger seam allowance - about 1/8" less than how wide you want your binding to be. 
Folded in half and pressed, our 2.5" binding strip is 1.25". We'll lose about ~1/4" covering the quilt sandwich and in the seam (the few threads that go over the thread!). That leaves us with 1", but we want a little bit less on the back and a little bit more on the front so that the front covers the line of stitching created when we attach the binding to the back. This means we'll set our seam allowance at ~ 3/8" to attach our binding to the back, and have around 5/8" on the front. 
Leaving about 10" of tail, line up the raw edges of the folded strip with the backside of the quilt. Begin sewing about 2/3 of the down one side, and backstitch when you begin. Go slowly making sure the raw edge of the folded binding is always lined up straight with the raw edge of the quilt top. You can pin or clip it in place as you go if it's easier for you.
Mitered Corners
8. However wide your seam allowance is, stop sewing that far from the corner (in this case, 3/8"). It helps to stop - needle down! - when you're a few inches away from the corner, measure the distance you need, and put a pin there so you don't sew past it. When you get to the pin, put your needle down, presser foot up, and pivot the quilt slightly. Sew off the quilt at a 45° angle, right to the corner as in the diagram below. Do NOT cut the thread.

9. Fold the binding strip up along the 45° stitch line, then fold it back down. Ensure that the binding strip, now along the next side of the quilt, is straight up and down along the raw edge of the quilt back. Make sure the fold you have just created is parallel to the first edge you sewed, as in the diagrams below.
(a)
      (b)
12. Begin sewing down the next side of the quilt, right from the very top of the quilt as in the picture. Repeat for each corner.

Joining the Ends
13. When you are back at the first edge of the quilt you started on, stop about 15" from where you began sewing. Backstitch and cut the thread. 
14. Lay the tail from the beginning of the strip along the quilt edge firmly as though you were sewing it down so there aren't any gaps later. Do the same with the end tail so that they touch. Finger crease both pieces where they meet. 

15. Unfold both tails so the strips are the full width (in this case, 2.5"). Twist the end tail up so the right side of the fabric is showing. Twist the beginning tail so that the right side of the fabric is face down. Cross the beginning tail over the end tail so right sides are together.
16. Now we need to line the fabrics up correctly. The right side of the beginning tail should be parallel against the finger crease on the end tail. The right side of the end tail should be parallel against the finger crease of the beginning tail. See the diagram below, I know this part is tricky. 

17. Sew from corner to corner of the overlapping pieces as in step 5 or the illustration above. Make sure the connected section of binding lays flat along the quilt back before you trim the excess strip tails! 
You may need to unpick and adjust if it does not lay flat. Make sure each piece is right up against the finger crease of the other, just to the left of the crease. If there's a gap (too much binding strip), you're too far to the left of the creases. If it's too tight (not enough binding strip), you've gone over the creases. 
18.  Once satisfied that the joined binding lays flat - not too tight or too loose - you can trim the remainders of the tails as in step 5, finger press open, then refold the binding back in half so both raw edges are lined up with the quilt edge again. Resume sewing where you left off. Remember to backstitch. Sew down the remainder of the binding and backstitch again.
Give yourself a minute or several. In fact, take as long as you need. 
Okay. Phew. Breathe out. You ready? This is the home stretch. Here we go.
19. Flip the binding up away from the quilt back. Optional: Press fabric up. I don't always do this because it's kind of tedious to maneuver the quilt so much but it does give a tidier finish. Flip the quilt over.
20. Wrap the binding around to the front of the quilt. Pick a side, halfway down. I like to start on a different side than where I began attaching the binding to the back just to be contrary. Hold the binding firmly in place. Not too tight or it will pucker, not too loose or there won't be enough batting inside the binding. We want the binding to be "full."
We also want to make sure that the front binding is covering the stitch line showing through from the seam on the back. You can pin or clip it in place as you go if it's easier for you.
21. You'll want to sew as close to the folded binding as you can consistently. For our example, recall that there should be around 5/8" of binding covering the front of the quilt. We'll give ourselves ~1/8" from the edge of the binding, so this should be a roughly 1/2" seam allowance from the edge of the quilt itself, and how we'll line the edge of the quilt up for our presser foot. If your binding width is different, you'll want to go back to your math for how much fabric will be on the front of the quilt and subtract 1/8 for what to set as your seam allowance.
Begin sewing. Backstitch. Stop a few inches from the corner (needle down). The binding should be straight out from the backing on the next edge and the binding along the edge you're currently sewing should be folded down over the quilt top. Note that when both edges are arranged like this it will form a triangle at the corner.
22. Now fold the binding on the next edge down over the quilt top and over the binding on the side you're currently sewing. This will create a 45° angle. You may need to gently adjust the fold and how it lies to get the angle accurate, and it's okay if it's not 45° and they don't meet evenly. Just do your best. Pin it down - ideally this will be the same distance from the edge of the next side as your current seam allowance, in this case 1/2".
 
23. As with the corners when doing the back, sew until you get to the pin. Stop with the needle down. Lift the presser foot and pivot the quilt so that the next edge is lined up for your seam allowance and you can continue to sew straight down along the binding, keeping it folded over the quilt top and covering the stitching line from the back seam. You should be ~1/8" from the fold of the binding as you were on the last edge. 
24. Continue sewing and repeat at every corner. When you get back around to where you started, sew just past the spot and then backstitch to finish. 
Ta-da! You've attached your binding! You may want to go back to the corners and stitch closed the folds for security. Cleo, Junie, and Ethel are proud of you! You should probably go drink some water now.