Here is a simple Dancing LED Circuit. The LEDs turns on/off alternately giving a dancing appearance. It is a simple Astable multivibrator using two NPN transistors.
It works on the principle of charging and discharging of capacitors C1 and C2. Current from the positive of battery flows through first set of LEDs D1-D3 to the collector of T1 through resistor R1. Resistor R1 limits current through the LEDs to protect them. The current through R1 charges capacitor C2. It then discharges through the base of T2 and resistor R4. This gives base current to T2 and it conducts. As a result second set of LEDs D4-D6 lights as the current flows through T1.Capacitor C2 again charges and the cycle repeats.
The same thing happens in the other side also. This gives alternate flashing of LEDs.