Diagonal weaving is a technique for producing bias-woven fabric on a pin loom. Bias-woven fabric drapes well and is ravel-resistant.

I'm using it here to make The Best Dishcloth In The World. Why the best?

Get dishes clean fast.
Wring out nearly dry for quick clean-ups.
You can bleach the heck out of them.
Nearly indestructible -- put them in your will.
Natural cotton -- I suppose they're bio-degradable.
Use a clean one to wring out the excess liquid from spinach -- it comes out nearly dry.
Dishwasher and microwave safe.
No enclosed pores to trap bacteria

I did eventually wear one out -- but I might have washed a chipped piece of glassware and cut it. It lasted for 3 years, though.

Step 1: Tools and Supplies

Very simple:

A pin loom of the desired size.
I've seen them up to shawl size, but for this demo I'm using 12x12 inch. They're easy to make: Pin Loom For Weaving Fabric Squares
Worsted weight 100% cotton yarn (one 4 oz. skein will make 3 dishcloths)
I plan to bleach these dishcloths, so I stick with white or colors that bleach to a pleasant shade. (For this demo, I'm using a natural color green that bleaches to a beautiful gold.)
A size 8 or H crochet hook
I file the handle end to a flat and use it as a beater. Some people use a table fork, but I like to use one tool for everything.
A tapestry needle
It sounds exotic but they're sold at almost all craft stores. It is just a blunt needle with an eye big enough for the worsted weight yarn.

Step 2: Start With a Slip Knot at the Top Corner.

Since you are making a diagonal weave, the loom should be oriented with corners to the top, right, left, and bottom. A slip knot is handy because later you'll want to run in the ends, and once it is off the pin, a simple tug takes the knot out of the yarn.

Step 3: First Pass

From your slip knot at the top, run the yarn around the pin at the bottom corner, then up to the first pin to the right of the top corner.
Wrap the yarn around the pin clockwise, and pull it under the strand that is knotted to the top corner. Since it is a loop, this also pulls the yarn under strand attached to the bottom corner.
Then put the yarn over the first pins (top and bottom) on the left side.
Take out the slack, but keep things loose. The strands will tighten up as you weave, and even more when you do the shrinking of the finished article.

Step 4: Second and Remaining Passes

Weave your crochet hook through the existing strands, across the newest strand, and pick up the free yarn.
Pull the yarn through to make a loop. (I find it helps to turn the crochet hook so that the hook opens toward you. If the hook is up or down, it tends to catch on the existing strands instead of sliding by them.
Put the top and bottom of the loop over the next pins on the left.
Tidy up the strands (see next step.)

Step 5: Let Your Tensions Go!

The goal here is to keep the weaving tidy, but not let it get too tight, especially during the first half of the weaving. As noted above, the article tends to tighten up on the loom as it is woven, and it will tighten even more in the shrinking step.

So, see the sag in the strands? That's about right.

You can maintain a nice. tidy weave while keeping the tension down. Just press down on the vertical strands while you are straightening the horizontal strand you just wove. This straightening is called "beating," but don't take that too literally. I just use the flattened end that I put on the crochet hook to tuck the horizontal strands straight. Some people use a dinner fork, but I don't like to switch tools.

Step 6: Optional: Color Change and Running in Ends

You can change colors by running both colors over and under a few (6 or so) strands together, then pulling the free ends down through the article and trimming them off. I don't care for the way it looks, so here's a way to make the transition over at the edge, where it is a bit less visible in the finished article.

Attach the new color over the old on a left-side pin, and then just weave on. That part is simple.

Running in the ends without making a dent in the edge requires a bit of planning, Basically, you have to make sure that the yarns are wrapped around each other before they're run in.

Running in is simple -- just go over/under 6 strands and trim the free end off under the article. I like to leave 3" until after the article is shrunk, then trim to fit.

Step 7: Finishing Touches

For the final pass, or anywhere adjacent strands are going over and under the same cross strands you'll need to run a single strand between them to lock the loop in place.

Note that I've over-beaten the top and bottom strands. It looks sloppy, but it's much easier to run the needle between them with a bit of clearance -- and the shrinking process will straighten everything out.

Actually running the single strand is pretty straightforward. Measure enough yarn off the ball to make a pass, plus 6 inches or so. Using your carpet needle, weave it down the middle of the final loop, over and under the opposite vertical yarns. When you get to the left end, wrap it around the end of the loop, and run in the end.

Double check that all the ends are run in, then trim them off leaving 2 - 3 inches. Wash and dry on high to even out the weave and shrink it tight. I like to add bleach, so that the dishcloth doesn't change color when you clean the sink with chlorine bleach, or wash/sterilize the dishcloth after use.

Finally, if you mess up, it's only a dishcloth, after all. If it comes out exactly as planned, well, it's the Best Dishcloth In The World, so it is worth a bit of extra effort.