Finger knitting is easy, tactile, and requires no equipment. A lot of young children (mine included, in years past) seem to enjoy seeing how quickly their piece grows. There are two methods that I know of.
Finger Knitting Method 1
Although the following method is sometimes called finger knitting, I call it finger crochet. It's a way to make a chain stitch without a hook.
First, the slipknot:
Once that is done, you continue to make a series of slipknots using your fingers. Just reach through the loop, grab yarn from the long end of the slipknot, and pull it back through. Continuing this, you create a chain:
Adding a crochet hook can feel a little unwieldy at first. It helps to remember to turn the hook up when catching the yarn, and down when pulling the yarn through.
Finger Knitting Method 2
This method is much closer to knitting as we think of it. You cast on by weaving the yarn around your fingers--first one way, and then back again. Once you have yarn on both sides of both fingers, you will bring the yarn straight across all four fingers as in the last picture of this sequence:
Now make your loops. Bring the yarn that is wound around your fingers up and over the yarn that is going across the fingers. It helps to bend your finger so the loop can just slip. Bottom yarn over top yarn, like so:
Continue with each finger in turn. Bring the bottom yarn over the top yarn for the pointer, the middle, the ring, and the pinkie fingers. Then you'll pass the yarn across the palm side of your fingers again and bring the bottom yarn over the top for the pinkie, the ring, the middle and the pointer fingers. Back and forth, forth and back. It goes pretty quickly:
In about a minute, you'll have a piece of knitting that looks like this:
Some years ago, one of my boys made scarves for his aunts using this method of finger knitting. They are narrow, but they knit up so quickly that it's possible to do a nice long scarf in a single sitting. If you need to, you can also transfer the loops from your fingers on to a large knitting needle until you're ready to pick up again.
Knitting is nothing more than making yarn into fabric by creating a series of loops, and the best demonstration of that is knitting directly on your own fingers and making the loops with your other hand.