Crochet may not be as old as knitting, but it certain has it's own claim to fame. Progenitor of many an intricate lace doily that graced your grandmother's home and that of fine ladies in the Regency era. It was all the rage in the 70s with "granny squares" and if you're of my generation you likely have curled up in a granny square afghan before.
Crochet, like knitting, at it's core is the simple twisting of fibers with a tool. The major pull and benefit to crochet over knitting for many is that it look less complicated with the use of 1 tool, rather than two simultaneously. The reality is that it isn't any easier or harder. While knitting has only 2 basic stitches, crochet has 5 (ie., slip stitch, single crochet, half double crochet, double crochet, triple crochet) basic stitches. There is only one way to decrease or increase a stitch however unlike knit which gives a veritable plethora. Crochet also has a number of special stitches from elongated versions of all of it's basic stitches to decorative and textured stitches (eg., popcorns, bobbles, puffs, picots...).
As with many fiber arts starting to crochet is as easy as acquiring hook and yarn, and of course a little instruction. Both allow for a myriad amount of choices to suit your taste, style, and wallet. Hooks are available in aluminum or steel, depending on the size, but also plastic, acrylic, and wood of various types. They also come in differing styles, including standard, Tunisian, cabled, and two differing types of double ended hooks (one version is for working Tunisian back and forth the other to combine hooks, they may face the same direction or opposite ones.) Yarn is available in the form of synthetic man-made fibers (eg., acrylic, nylon, rayon), plant-based fibers (eg., cotton, linen), and animal-based fibers (eg., wool, mohair, angora, cashmere, alpaca, silk) or a blending of any of these. Due to the potential size of a project crochet has an easy portability to it, except in the case of larger pieces, such as afghans.
Crochet differs largely in some ways from knit in it's basic use. Crochet is perfect for sculptural and three dimensional objects. (This is not to say that you can't do such thing in knit at all, just that it is easier to do so with crochet.) Unlike knitting crochet has a denser feel to it and is worked up one stitch at a time. The benefit of this is it's great for outerwear and things that need to be heavier or thicker, and if you do have a hook mishap and drop your work you likely only messed up one stitch. (As opposed to the catastrophic dropping of needles where in an entire row could unravel!) With the greater amount of density however you use roughly 3x the amount of yarn. So a similar project would take more yarn in crochet than it would to do in knit.
Crochet comes in a number of styles, including amigurumi (the making of small dolls or toys comprised entirely of single crochet, originally made in knitting), Irish Crochet (a particular style of crocheted lace), Tunisian Crochet (uses a special kind of hook that is either elongated, has a cable to extend it, or is double ended), and Freeform Crochet (this technique involves a number of crochet techniques that build upon one another to form often abstract designs).
When choosing to learn crochet depending on your style of learning there are many options. YouTube offers videos on everything you would need to get started. Many sites also offer full color, live picture tutorials so you can "see" stills of how a stitch started, in progress, and finished should look. There are a plethora of books and magazines about crochet, that all cover the basic stitches.