Maybe it's just me, but when I see "M1" written in a pattern, I think of English motorways, like the M5. Maybe I've just read too many novels set in England. Anyhow, the M1, or "make one" increase, is very useful, in part because it is actually two increases -- the increase can be made to slant to the left or to the right, usually abbreviated "M1L" or "M1R." Because the increase can be slanted, shaping can become a design element in your knitting, with the slants of increases and decreases playing off one another in elegant symmetry.
To work this increase, you will need to find the bar. Not the cold-beer-on-tap kind of bar -- you're looking for the bar of yarn that connects your stitches. If you hold your needles a little bit apart, you can see it, as in the photo below. Dip your left-hand needle down and slip it under the bar...
...like so. The bar of yarn is now nearly ready to become a new stitch. Note the orientation of the bar: the strand of yarn moves from the back over to the front. So, you'll want to knit it through the front, as you would an ordinary knit stitch in order to twist the strand closed, preventing a hole.
Here, I've placed the arrow so that it is parallel with the twist of the new stitch -- you can see that it slants to the right. If I were increasing for the bust of a sweater, I'd probably use this increase on the right-hand edge of the garment, and a left-leaning increase on the left-hand edge, to echo the line of the fabric itself.
Speaking of left-leaning increases...If you want your make one increase to slant to the left, place the bar of yarn so that the strand goes from the front of the work to the back, as seen in the photo below.
Here you can see that the stitch slants to the left, and so is an M1L. In the swatch below, the M1L and M1R are bookending a single plain knit stitch so that you can see the differences between them. The knit stitch in the middle has no slant -- its legs simply aim downward in the "vee" shape typical of all regular knit stitches -- while the others have a clear slant.
Aside from using this increase for waist and shoulder shaping, they are also ideal for thumb gussets on mittens, or anyplace where you need a few paired invisible increases. The main drawback of this type of increase is that it eats yarn from the row below -- so you would not want to work many M1 increases in the same row, or concentrated in a small area, or your fabric will pucker.
Don't worry if you can't remember which way to orient your yarn to twist to the left or right; it's easy to slip the bar of yarn back off the needle and try again until it looks right.