In no particular order...

Thought I'd start with this, as there's already a photo of it here.
Basically, at the request of mum, I made this (she asked for one, but didn't specifically ask me to make one). It required:
A sewing machine (or needle and thread, if you're not the machine type). For the record, mine is a singer sewing machine. Class no. 201K. It was a present to my granny on the occasion of her 21st birthday, which makes it an impressive 73 years old this year. And yes, it's electric.
Some fabric (slightly smaller than a tea towel)
Some elastic (I butchered a pair of boxer shorts, (sorry mum!) but I gather that the more conventional method is to buy the stuff). Two bits of elastic, to be precise.
Some pins (you can decide for yourself how many you need)
A bit of ribbon to hang it up with

Basically, you want the elastic to be about 1/2 to 2/3 as long as your fabric is wide (make sense?) e.g., if your fabric is 30cm/12in, you want the elastic to be about 15-20cm/6-8in. You need it to create a hole that is the right size to allow plastic bags to pass through, without being so big as to let them all drop out!
Pin your elastic to the top and bottom (the shorter sides, that is) of your bit of fabric. You probably want a pin at each end and one in the middle.
Sew the elastic to the fabric using the machine. To be honest, this was a bit of a faff, but I found that the 'grip' (sure there's a technical term for it) on the sewing machine was good enough for me to put the foot down on one end of the elastic and fabric, and then pull the elastic taut from the other end whilst sewing along the length. Do this for both bits of elastic!
Hem the ends (also tricky because of the elastic) by folding the fabric over the elastic and then fold again so both the elastic and then end of the fabric are hidden, and then sew along.
Sew the two sides together with a straight stitch, making sure that you have the fabric inside out, so the seam ends up on the inside.
Turn the whole thing the right way around.
Sew the ribbon into the inside of the top (the top is whichever end you want it to be...) I think I did this by hand, as I thought if would be easier than a machine.
Hang it up and marvel at your creation. (Trying to avoid looking at bits that don't quite match up, and wondering why you used yellow thread on a green and white fabric)

Good fun to make, satisfying too. And 'made to order'. Must ask mum for a report on its progress and functionality...

Simples.