We've all seen intricate things made using crochet, and most folks find the number of different crochet stitches possible to be a bit overwhelming. Chain stitch, single crochet, double crochet, half-double crochet, shell stitch, filet crochet, tunisian crochet, puff stitch, popcorn stitch... How daunting!
When taken at its most basic, though, in order to do any of those stitches, one only needs to know how to do three things: insert hook, yarn over, and pull through. What differentiates the stitches is how many times and in what order you do those three things.
To do a chain stitch, which you pretty much start every crochet project with, you only need to know two of the main three maneuvers: yarn over, pull through.
To start, make a
slip knot in your
yarn and put it on your
hook.
Hold the yarn in a manner you find suitable, and hold your hook either like a knife (my preference), or like a pencil (I find this hard on my wrist, your mileage may vary).
To do a yarn over (yo) start with the hook above the yarn:

Rotate the hook around the yarn in a counter-clockwise direction:

That's it! You'll do a LOT of yarn overs in your crocheting career.
The next step in a chain stitch is the pull through. Start with the yarn over, then continue like this:
Look at the loop that is on the hook already. You'll see that it is not a circle shape, as one might expect, but a tear-drop shape. Turn your hook so that the end of the hook is lined up with the pointy part of the tear-drop:

Making sure you've caught the yarn in the hook, pull it through the loop on the hook:

Push the hook up so that the new loop you just made is on the shaft (the part of the hook between the hooked end and the flat thumb rest). This is important to getting a nice, even, easy-to-use chain:

And that's it! Do that over and over again to produce a "base chain" which is the starting point of a flat piece, or to make a starting loop for crochet in the round (tutorial to come!).

Things to do with the miles of chain stitches you're likely to produce while practicing this:
wrist wrap,
scarf.
Good luck!