First of all, you'll need to know some basic circuitry principles and you will need:
A battery (9V is a good choice for a large selection of LEDs)
A collection of wires
Some splicing caps (or alligator clamps attached to your wires)
A resistor of the correct resistance
Some LEDs.
The resistance for the resistor is a bit of a ball park thing - anything less than a full order of magnitude (10x or 0.1x) shouldn't cause a significant difference.
Ideal resistance can be calculated using a number of parameters:
Battery Voltage
Voltage drop across LEDs (red and green are 2, blue and white are 3.5 (and blindingly bright - they're meant for illumination, not decoration)
desired current (about 20mA for most cases).
Number of LEDs to connect
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