Step 1: Materials
Just about anything that is vaguely fiberlike can be spun. The most common fibers used for handspinning are sheep's wool, cotton, silk, alpaca, mohair (from angora goats), and angora (from angora rabbits). In my opinion, the easiest to learn to spin is sheep's wool, although the principles are the same for spinning any fiber.
You'll want to work with prepared sheep's wool, known as roving, which has been washed to remove excess lanolin (grease) and carded or combed to orient all the individual fibers in the same direction. You can find roving at your friendly local yarn store, or from several online retailers. I like to peruse etsy.com for handpainted roving, myself. Try to avoid "top" for now, which is wool that has been combed to leave only the longest fibers, which makes things more difficult for the beginner, although it is delightful to spin with a little practice under your belt!
The wool pictured in this instructable is a Romney and Merino blend (these are breeds of sheep.)
At it's simplest, a spindle is really just a stick with a weight on it somewhere (aka a whorl.) Drop spindles come in two main flavors: top whorl, and bottom whorl, cleverly named for the location of the whorl on the spindle. Spindles also come in many different weights, but I recommend starting with a spindle that is around 2-3 ounces.
The spindle pictured is a top whorl 1.5 oz spindle, although you could also use a bottom whorl, or even the spindle that stores your CDRs. Here is an instructable that will show you how to make a spindle from a CDs, a dowel, and a hook: http://www.instructables.com/id/Drop_Spindle_Constrution. You can also find spindles from online retailers or at your local yarn store.
Step 2: Pre-draft
Keep gently stretching out the roving until it is at least twice as long as the original length.
Step 3: Fiber management
This is essentially serving the same purpose as a distaff, although for our purposes and small amount of fiber, a wrist will do just fine.
Step 4: Attach a leader
Before introducing any fiber into the equation, you can give your spindle a (clockwise) spin by twirling the shaft below the whorl and letting it hang from the leader. Look how the spindle keeps going for some time before the twist in the yarn fights back and slows it down. High Tech!
Step 5: Join fiber to leader
Step 6: Park and draft
Bring your right hand up to meet your left, and pinch where the twist ends, freeing up your left hand to move back. The fiber between your hands is called the "drafting triangle." The amount of fiber in the drafting triangle will determine the thickness of the finished yarn. If you want a thinner yarn, draft out the fiber in the triangle more. When you are satisfied with the thickness of the drafting triangle, release your front (right) hand, letting the twist run up into the triangle, and making yarn!
Repeat this process, moving the front hand up to the new end of the twisted section, moving your left hand back, drafting, and letting twist into the newly drafted fiber. When you want to add more twist, hold the yarn with your left hand at the bottom of the drafting triangle, where fiber turns into yarn - it is important to keep a leash on that twist. Use your free hand to spin the spindle, building up more twist, and park it again. Keep going until you have a foot or two of yarn, then go on to the next step.
Tip: If you find there is too much twist in your drafting triangle to comfortably draft, try untwisting by rolling the fiber in your right hand. You might also find that you do not have enough twist, and it feels like your yarn could easily be pulled apart, in which case you'll want to add more twist.
Don't worry about how it looks at this point, worry more about how the process feels, and just let your hands do a lot of the understanding.
Step 7: Wind on
Step 8: Add more fiber
Tease out a few fibers, hold them together with the fiber at then end of your handspun, and add twist until the fibers grip together. Twist is like glue for fibers, and they will magically grip onto each other. Now continue spinning as before.
Step 9: Take off the training wheels
Keep spinning until you have a full spindle!
Step 10: Wind a hank and block
You could invest in a niddy-noddy, which is specifically designed for winding yarn into hanks, and is fun to say, but you can also just use your forearm. Wind the yarn off of the spindle and into a series of loops around your thumb and elbow. Tie the ends together, and use scrap yarn to make one or two ties around one side of the hank, which will keep the yarn from getting tangled. Your hank will be all wonky and squiggly. Do not fret.
To block your yarn, submerse it in lukewarm water, gently squeeze out the excess water, snap it between your hands a few times (or some folks even whack it against a hard surface) and hang it for drying. I hang my hanks on doorknobs, or shower curtain hooks. Since this is a single, and has a lot of active twist, you will also want to weigh it down it to help get the kinks out. A spray bottle full of water or other liquid works nicely as a weight, since you can easily hook it onto lower half of the hank. Alternatively, if you happen to possess a yarn swift, you could stretch your hank on that to dry.
You can turn a hank of yarn into a skein by twisting it a few times, folding it in half, and pulling one end through the loop at the other end.
Step 11: Practice practice
Most importantly, make things with your handspun! In the words of my spinning teacher, Maggie Casey, a spinner who doesn't use their yarn is like a baker who doesn't taste their bread. Knitting, crocheting, or weaving with your handspun will give you valuable feedback, and help you make informed choices during the spinning process.
Spin spin spin!