The instructable uses a 3PDT switch. That's three pole, double throw.
Each pole switches one connection. You need three poles because you want to switch all three connections on the jack socket - tip (left channel audio), ring (right channel audio) and sleeve (ground).
The number of "throws" (positions) determines how many devices you can switch between. The original design, with a double throw switch, switches between two devices. (The centre position is OFF, nothing connected).
I'm assuming that you want to switch between three devices, or four devices. In that case, you will need a 3P3T or 3P4T switch.
Your best option is a rotary switch. These are available with lots of poles, and lots of positions. They can be quite expensive though. Here are some options from Digikey.
Their cheapest three pole, four position rotary switch is USD 5.50:
http://www.digikey.com/product-detail/en/KC34A30.001NLS/EG1956-ND/101803
This one can be set for 2, 3 or 4 positions but is more expensive (USD 11.04) (it's also much better quality):
http://www.digikey.com/product-detail/en/C5P0304N-A/451-1029-ND/514083
Both of those switches are non-shorting types. Digikey have other options too; those are only the cheapest alternatives. Rotary switches are available from other suppliers too.
If you have the choice between shorting and non-shorting, go for non-shorting. A shorting switch will briefly short sources together as you change its position. This is not likely to damage anything though, if you are just switching line-level or headphone-level signals.
Switch boxes are available ready-made, although they usually use RCA Phono connectors instead of 3.5 mm jack connectors.
If you actually want four-pole switches, a rotary switch may still be your best option, but you can find four-pole switches of other types as well from online suppliers such as Digikey and Mouser.
