All you need is a strong cardboard box and some wool. Cut the top flaps off the box and make some notches down each side about half a centimetre apart. The size of your finished weaving will be determined by the width of your box and the number of notches that you make. Then thread the wool between the notches to form the warp threads. Knot firmly to secure. Use some of the cardboard from the box flaps to make a comb for pushing down the wool and a shuttle to thread the wool through.
Then choose the wool that you would like to use for weaving. It's good if you can find a selection of different colours and textures to keep your child's interest, and you can also use ribbon which will make the process quicker as it is thicker. Tie firmly to the shuttle and weave in and out, leaving a thread hanging out at the beginning. Make sure that you don't pull the weaving too tight, and every few rows use the comb to push the threads neatly together.
When you have finished you can either sew in all the loose ends, or just knot them together and cut them off. Once knotted you can also leave them there to make a fringe. It's best to start small so that your child is able to complete a whole piece in one go. Younger children will need some supervision to make sure that they remember to alternate going over and under between the rows and don't get in a tangle, but older children will soon be able to manage by themselves.
The finished piece of weaving can be used as a bookmark, or as a little mat for a dolls house or play set. If you really get into it you could turn them into gifts for adoring relatives like mats, or sew them in half to make phone or glasses cases. You could also weave in beads for some extra texture. It's a great activity for developing fine motor skills, and also for learning a little bit about how fabric and clothes are made.
If you look on YouTube there are pleny of videos showing weaving in action that you can watch with your child, both mechanical and by hand. We watched a few and I found this one particularly good as the loom is just a slightly advanced version of the one that we made: