A double-pointed knitting needle is a knitting needle with points on both ends (as opposed to a standard knitting needle, which has some kind of stopper on one end). They are sold in packs of four or five and are used for knitting projects in the round when a circular knitting needle would be too large.
Double-pointed knitting needles are necessary to make socks and mittens or gloves. You may also start out with circular needles and then switch to double-pointed needles when a project needs to be made smaller — when knitting up the top of a hat, for instance.
How to cast on using double-pointed needles
Begin by casting on the specified number of stitches onto the specified number of needles. A pattern will let you know how many to use.
You would then slip half of the stitches onto the second needle, then repeat this for each subsequent needle until each has roughly the same number of stitches. (When dealing with an odd number of stitches, one needle will have more than the others.)
The needles should cross over each other to form a triangle, in the case of three needles, or a square, in the case of four.
Place a stitch marker between the stitch with the working yarn and the next stitch and join the yarn for knitting in the round as you would with circular needles, being just as careful to avoid twisting the stitches so as to avoid creating a möbius strip.
The craft website Purl Bee helpfully suggests painting one end of each of your double-pointed needles a different colour; this helps distinguish them when you're casting on.
Double-pointed needles are often abbreviated as "dpn" in knitting circles (no pun intended). They can be difficult to adjust to at first; I have never sworn so much out loud as I did the first time I tried to knit a small tube using them. Like other knitting needles, they come in different materials — metal, plastic and wood. You can also purchase cubic double-pointed needles, which are apparently easier to hold.
References
Knitting in the round on double pointed needles
Double pointed needles