An easy way to weave with construction paper
The best way to understand weaving is to weave something for yourself. You can do this with just strips of paper: take a piece of yellow construction paper and cut it into inch-wide strips. Cut a piece of green paper the same way. Lay all the yellow strips side by side and tape them to your table or desk with one long piece of tape across the top. Weave the green strips through the yellow ones, being sure to go over, then under, then over, then under. The second strip should reverse this: go under, then over, then over. When you've got your weaving as big as you want it, use tape to secure the edges, and you've got a little mat.
To weave with yarn, a good way is with a piece of cardboard as in the video. You can also use one of those potholder kits, and then you end up with a useful potholder for the kitchen when you are done!
Weaving on a cardboard loom
But this simple weaving isn't really how people made clothes - real looms have a heddle so you don't have to actually go over and under each thread. A heddle lifts every other thread up, all at the same time, and makes your weaving go much faster. The second loom listed at the bottom of this page has a heddle, so if you want to see what it's like to weave with a heddle, that's the one you would want to get. A heddle is hard (though not impossible) to make for yourself.
More weaving projects and books about weaving: