Technically, fletchings are the “feather bits” at the end of an arrow which help it stabilize during flight. Fletching is the act of affixing these fletchings to the arrow, though if you ever actually make a fletcher, it would be a person that makes arrows (and probably even the odd bow) from start to finish. Note that someone who specializes in making bows is a bowyer.
Arrows need only a few things in principle to work; a sharp and heavy head to punch through the target, a straight and narrow shaft to drive and deliver that motion, and finally fletchings for drag and stability.
Take into account that an arrow can be crafted out of nearly any material so long as you follow the basic principles of what is required. Whether your bow will be strong enough to fire it accurately will be another question. For beginners, go traditional; find sticks. A good rule of thumb is to use something straight (obviously), and slightly thicker than a pencil. Look for long sticks or branches that are light and sturdy. The longer the better, as you can always size arrows down if need be. Alternately you can buy arrow shafts or complete arrows. Easy.
Next (if you’re not buying), you will need a head for the arrow. Also something easy to purchase in variable forms, but for the do it yourself enthusiast or apocalypse survivor, you will want to find something with a bit of weight and slightly wider than the arrow shaft. Ideally, this is a pointy and funneled or bladed piece made out of rock, shell, bone, glass or metal. Commercial arrow heads essentially screw right on, while a self-made tip will need to be affixed with glue, string, or ideally both (you can also use fletching tape in a pinch). Cut a notch into the front of the shaft to nest the back of the arrowhead into. Use some sort of adhesive glue to bond the two items, then wrap it securely with whatever kind of string you can find. Add another layer of glue or adhesive along with the string to strengthen it. Allow this bond to dry for a couple hours by laying the arrow flat in a position that does not rest weight onto the arrowhead. Turn it over after an hour or so to make sure the glue dries and hardens evenly. Place it out in the sun or under a heat source to speed the process.
Next you will want to square off and cut a notch in the back of the arrow shaft. This is where your bow’s string will fit into to aid in holding the arrow and keeping it steady. That said, this notch does not have to be very deep and not much wider than your bowstring itself.
Finally comes the actual fletching. Traditionally we use bird feathers, split down the middle and arranged at the end of the arrow’s shaft to promote straight or spiral flight. Each has it’s use and experimentation will yield what works for you. For straight flight, the fletchings are aligned straight with the length of the shaft so that it does not curve in the air. For a spinning arrow, the fletchings are slightly curved in the same direction to generate a spinning motion through the air. It’s all about that gyroscopic stability. You can use 3 or 4 fletchings, and different colored materials can be used to mark different kinds of arrows for easy identification; like arrows with different heads or with curved or non curved fletchings.
No matter what material you use (plastic, paper, feather or metal), the fletchings must be secured to the shaft well enough so that they do not fall off during transportation of the arrow or during it’s flight. Fletching jigs are available commercially and on this site to make easy work of the actual fletching process. Otherwise you will have to stick to adhesive glue or tape and careful precision when laying your fletchings down. Alternately you can also acquire a simple ready to stick on fletching. Feather fletchings can be separated in gaps to create slots where the fletching can be secured with string, though it is still recommended to use some form of glue in addition to this to make it stronger.
If you are making arrows from scratch, the best thing you can do is practice. Create two or three variations based on the materials you have at your disposal and discover what will work best for your intended purpose. You may notice subtle adjustments to be made that can increase efficiency and stability. There is no single right way to make or fletch an arrow.
Just be warned; arrows are dangerous even when made from modest materials and fired from simple bows. This is not an instrument of peace.