This is a fairly simple project for anyone with some electronic skills.
Plug this circuit into an audio source and the LEDs will blink to the rhythm of the music. All it is really is using the voltage from the audio input to activate a transistor and turn on the LEDs.
This is circuit is very customizable. The setup of the lights is really up to you. The main thing to consider when choosing layout for the LEDs is your audio source and the power supply you will be using.
For this particular one I choose to make it run off a 9v battery and use an IPod for input.

Step 1: What You Need

Depending on the configuration of you setup you may need to modify the number of parts below.
Parts:
Superbright LEDs, the number and color is up to you (I used 4 blue ones).
2 x TIP 31 NPN Transistor, the heart of the project.
Audio cord (a headphone cord will do or just an 1/8 inch jack)
case (I used a spare thick cardboard box)
9v battery clip (or the power supply you are using)

Tools:
Soldering iron
Wire cutters/Strippers

Step 2: Hooking Up the LEDs and Transitors

1.Solder 2 of the LEDs together in series. Connect the anode (long pin) of the first LED to the cathode (Flat side/short pin) of the second LED.
***NOTE you can add more LEDs in series or a resistor depending on the voltage of the power supply.
2. Now connect the Cathode of the first LED to the middle pin of the TIP 31 (the collector).
3. Repeat steps 1-2 again (for the second part of the stereo input)
4. The Anodes of the last LEDs go to positive voltage.
5. The right pins of the TIP 31s (emitter) go to ground.

Step 3: Hooking Up the Audio Input

Take the audio cable, strip the ends, and determine which wire is right and which is left. Red is usually right and white is usually left.
Another way to determine the sides is to take a multimeter and do a connectivity test with each wire and part of the jack
The tip of the jack is right.
The middle of the jack is left.
And the bottom of the jack is ground.
NOTE****
Decide which set of LEDs will be right and which will be left.
Then connect the corresponding wires to the left pin of the TIP 31s

Step 4: Final Assembly

Connect the positive part of the battery clip to one side of a switch and the positive ends of the LEDs to the other.
Then mount it all in a case (i used an old jewelry case).
I used a hot gluegun to secure everything, and some electrical tape to protect against any shorting.

Step 5: Testing and Modifying

Plug the jack into an audio source and turn it up until you see the lights. Some players will not be powerful enough to drive the lights. I suggest any song with a distinctive beat. (I used Crazy Train)
If you want to listen to the music as well you need to get a splitter.
Now its your turn. This is just one of many ways to wire up this circuit. Good luck and be creative.
Thanks to the following for some of the original ideas:
http://www.danielandrade.net/2005/12/06/howto-blinking-leds/
https://www.instructables.com/id/LED-Music-Sync-Lamp/
This is my first instructable and i would appreciate any feedback.

Step 6: Updates

Here are some additions/modifications that people have been suggesting and asking for.
1. The first addition is adding an output jack wired in parallel to the audio input wire. Unfortunately the case I used was a little too small for this.
2. Different configurations of LEDs. Many people commented on using different numbers/colors of LED. The main thing to be careful of is VOLTAGE that you draw. (Series vs parallel). The next step I am going to try is maybe using a RGB LED.
3. Transistors. I am not an expert at transistors. Use a NPN that is rated for the appropriate voltage.
4. Mounting the LEDs in interesting cases. I am going to try mounting them in a pair of portable speakers.
Thanks for the comments and good luck!