This is a paper airplane of my own design. It is small, and will fly as fast as you can throw it. Its only downside is that it tends to go into a spin when thrown across the wind. Being so small and fast, it's very sensitive to trim, so don't overdo it.
You will need paper, scissors, and tape.
Step 1: Folding
Start with a half sheet of letter-size paper, folded in half width-wise. (The short way.) Fold the edges into the middle, to make a point. Fold the new edges in again, as if you were making a standard dart.
Step 2: More Folding
Unfold your last folds, and mark the center of your paper with a short crease. Don't make it too big, but make sure you can see it. Then, fold in one of your sides the way you did in Step One. Fold it back out again, starting at the center crease. (That was probably really confusing; just look at the picture.) Do it again on the other side.
Step 3: Even More Folding
Fold the inside floppy edge things (that's a technical term) in half, along the line I've marked in the picture. One side has the fold marked, the other side shows the completed fold.
Step 4: Even More Folding
Pointy noses look cool on airplanes, but we want ours to be balanced. Remember how you marked the point halfway between the front and back of the plane with a little crease? Fold the point of the nose down to that mark, then fold the new nose down to the point. I hope that made sense.
Step 5: You're Not Done Folding Yet
Fold the plane in half along your original center line. Then fold the wing/triangle thing down and crease it well. the picture only shows this on one side, but do it on both. When you've done that, leave the triangles folded down but fold the wings up, as shown in the picture.
Step 6: Make a Tail!
Our plane would look ridiculous with that huge triangle of paper hanging out the bottom, so we're going to use a reverse fold to make it into a vertical stabilizer/rudder. Fold it up on one side. There is no reference point for this fold, so you will have to judge what looks right. Unfold it, flip the plane over, and fold it the other way, preparing it for the reverse fold. Pull the wings apart a little and pop the tail up through the fuselage. You are now done folding!
Step 7: Tape
You origami purists may be cringing as you read this, but this plane needs tape to hold it together. Take a piece a few inches long and wrap it snugly around the nose of the plane, making sure not to warp it. Then, take a tiny little piece of tape and tape the tow sides together, under the rudder.
Step 8: Finished!
You will notice that each wing has two layers. Stick your finger between them and "poof" them out a bit, creating airfoils of sorts. Your plane is now ready to fly. (For whatever reason, these pictures seem to be upside-down. Sorry about that.)