Using some resistors and capacitors, I basically configured a 555 timer chip to act as an oscillator (fancy term for something that generates electrical pulses at regular intervals). I used the pulses to drive an LED, which turned on whenever there was a pulse and turned off whenever there wasn't a pulse (thus the blinking action). In technical terms, this circuit is called an "astable multivibrator."
This is a great project to start off with if you're thinking about doing electronics. Here's what you'll need:
-U1: 555 Timer IC
-R1: 1 megohm potentiometer
-R2: 47 kilohm resistor
-R3: 330 ohm resistor
-C1: 1 µF electrolytic capacitor
-C2: 0.1 µF (100nF) disc capacitor
-LED: 5mm red LED
Just assemble the circuit according to the attached schematic diagram (555 LED flasher.pdf), and you're set. For a nice explanation of how this actually works, visit the Electronics Club's 555 Timer page here.
Note: A month ago, I made an improvement to my original design that allows me to drive two LEDs instead of just one. Basically, the LEDs are connected so that whenever there is a pulse from the 555 timer, the red LED turns on and the green LED turns off, and whenever there isn't a pulse, the green LED turns on and the red LED turns off. I have included the corresponding circuit diagram (LED Flasher 2) with this Instructable.