I have always been interested in audio and music of all kinds. Recently, I tried my hand at building a speaker, and I was quite pleased by the results. Chances are you have most of the materials at your house, and it only takes about half an hour. There are two ways to do this, but one of them requires you either a) have an audio amp circuit already or b) remove one from a pair of computer speakers; I'll elaborate later.
A video of the speaker in action is available here: Speaker Demonstration

Materials:
Folder or thick cardstock (big enough for the size of the cone)
Magnet wire (30-34 gauge enameled) (If you take apart a small motor, the coils that make up the armature are made of this and can be reused)
Neodymium magnet (other magnets should work, but neodymium is the most suited to the job.)
Computer speakers with headphone output (or an audio amplifier circuit)
Aux cable (if you do not have an audio amp circuit you have to be willing to cut this)
Solder
Cardboard (for the base)

Tools:
Scissors
Compass
Soldering iron
Ipod/mp3 player
Lighter
Glue gun
Exacto Knife
Wire strippers

Step 1: Theory

For those who are unfamiliar: a speaker is based on the idea of electromagnetism; pass a current through a coil of wire and create a magnetic field. by placing a permanent magnet in the middle of this coil, and the active wire will be attracted to it. If the current that is passed through the coil has the form of an audio signal, the coil will move up and down to match the wave. When this coil is attached to a cone, also called a diaphram, the cone will create vibrations in the air. These vibrations are sound waves.
The output of an mp3 player is not powerful enough to create a strong field, so the amplifier circuit will boost the output to usable levels.

Parts Of a Speaker

Step 2: Making the Cone

The cone is simply a concave piece of paper. Take your folder and draw a circle on it with the compass. Draw a line from the center of the circle to the edge. Carefully cut out the circle, and the line you drew. Carefully pull one side of the split circle towards the other side; the paper will form a con on its own. Try not to crease or bend the paper. when you have a cone that you like, tape the edges.

Step 3: Make the Voice Coil

I apologize for this part, it is not easy to explain in words. Please refer to the pictures for clarification. The voice coil has to be slightly larger than the magnet. To accomplish this, cut a strip of index card lengthwise into two roughly 1.5 inch strips. Take the first one, tape it to the magnet, and wrap it around all the way. tape it down. Take the second strip and wrap it around. Once you have a cylinder, mark the place where the paper crosses itself. Remove the strip and cut slightly past the marking. Rewrap the strip and tape it to itself. This cylinder will be the voice coil.
The easiest way to wrap the wire around the coil is to get a drill bit the same size as your magnet, wrap masking tape around the bit until the cylinder fits snugly. Tape the edges of the cylinder down to the bit. Get your magnet wire and hot glue gun ready. Leaving about 5 inches hanging off the edge; glue the wire to the coil, near one of the edges. SLowly turn the drill on, letting the wire coil around. You only need about 1/2 to 3/4 of an inch of coil. Try to keep each turn tight against the one before it. A neat coil will sound better. When you reach the bottom, put a dab of glue to hold the last coil in place. Then, only using one or two turns, bring the end of the wire back to the top so that it is parallel to the other end. Put another dab of glue to hold this end in place. Cut the wire about 5 inches from the top of the coil.
At each end of the wire (about half an inch), scrape off a little bit of enamel with the Xacto knife. Use the lighter to burn off the remaining enamel from where you scraped.
Trim the bottom of the coil down so that you don't have a lot of excess paper (see picture).
Glue the top of the coil to the middle of the convex side of the cone.

Step 4: Set Up the Speaker

Cut four about 1 inch by 5 inch strips out of the folder. Bend each in half, and then bend in the ends about 1/4 of an inch (see picture). These will be supports. glue them at equal distances around the cone (see pictures) and then to the cardboard base. Make sure that the cone is as level as you can make it (if it is slightly off don't worry, you can adjust it by bending the supports more).
Take your magnet, and find a way to raise it so that it is supported directly under or in the coil (I used balsa wood strips). Glue the magnet and the support down to the base (see picture).

Step 5: Attach the Audio Source; Method 1

If you want to use an amplifying circuit, skip this step.
If you want to use computer speakers, take them and you aux cable out. Cut one end off (you could use headphones here, if you want.) Strip the end off; there will be three wires inside, a gold (or silver), a blue, and a red, or some variation. Gold or silver will be ground, red and blue are left and right channels. Take the red one and roll it in your finger to fray it. Repeat with the gold. There will be thin enameled wire and nylon wrapped together; you need to separate these. Then take the lighter and carefully burn off the nylon and enamel on the wire. Move the third wire out of the way. Take one of the two burned wires and solder it to one of the end of the coil wire. Do the same with the other one. Plug the computer speakers into the wall.

Step 6: Attach the Audio Source; Method 2

Take your amplifier board and locate which wires are outputs. My board is taken from an old pair of speakers, so has two channels. If this is the case for yours, you can usually combine the corresponding wires for each channel to each other (see picture). If you only have a single channel on your amplifier, don't worry about that. Attach the outputs to the wires coming off of the voice coil. Plug in the amp, turn it on, and attach the aux cable to the amp (See picture)

Step 7: Finish

Plug in all of your components (don't forget to connect your mp3 player to the aux jack), make sure your voice coil is centered on the magnet (see video), and turn on the amp/computer speakers. Start the volume off low, and increase it slowly. If you are using computer speakers, make sure you don't forget to turn the volume up on both the speakers and the mp3 player. If the sound is quiet, make adjustments to the position of the cone until it clears up. If you have questions or issues, post them in the comments and I will respond as best I can.
Here is the video again:
Speaker Demonstration