I currently have a Canon 60D and a Yongnuo YN-560 II Speedlight Flash, but am wanting to fire two flashes on tripods. Can I buy another 560 II to do this and how so? And if I can't then do I need to buy two 568's or...and what trigger will I need? (Needy, I know. LOL) There are a lot of similar questions in here but I didn't see exactly what I was looking for.
Your YN-560 II includes a simple optical S1 or S2 slave mode. You can use your 60D built-in flash to trigger any Yongnuo flash this way.
S1 fires whenever it sees any flash of light
S2 ignores the metering ETTL pre-flash and fires with the second main flash
If you use manual flash on your 60D, use S1.
If you use ETTL flash on your 60D, use S2.
S1 / S2 modes
Yes, you can purchase another YN-560II and use the S1 or S2 "dumb" optical modes, and your 60D's pop-up flash—as a regular flash, not as a Canon wireless master—as the trigger to set them off. You use S1 if the pop-up is in M mode; S2 if it's in eTTL so it will ignore the ttl preflash.
But this is going to be line-of-sight limited and could have issues with range and reliability if used outdoors in bright sunlight. Optical systems suffer these issues because the sensor has to "see" the triggering flash burst, and a pop-up flash is fixed (i.e., tough to use a flash behind the camera this way).
You have a number of other choices open to you when it comes to cheap Yongnuo gear, if that's the way you want to go, but all of the methods below would preclude using the YN-560II you already have, and would require that your purchase two flashes to get two off-camera lights.
Because the 60D's pop-up flash also has a built in Canon wireless eTTL master, you could purchase any of the Yongnuo flashes that has a name ending with an EX, and you can use it as a slave in Canon's optical wireless system. You'll have the same issues as with the S1/S2 modes, but you'll also have more features—eTTL, HSS, remote power control, etc.
The YN-560II you have, however, won't work in these scheme, and can't be used simultaneously in S1/S2 mode (wireless eTTL has multiple preflashes and would set it off early), so you'd have to purchase two flashes.
You could also get the upgraded YN-560 models with a RF-603/-605 radio receiver built in: either the YN-560 III or IV, or the upcoming YN-660, and then use a YN-560-TX dedicated radio transmitter on your camera hotshoe. This will give you radio reliability and range, free you from line-of-sight requirements, and still give you remote power and zoom control (although not TTL or HSS).
You could use the YN-560II in this scheme by adding on an RF-602/603/605 trigger, but you would only be able to fire it remotely, power/zoom would still have to be set on the flash itself.
Or, if you have a bit more to spend, you could get YN-685s, which have a 622 receiver built in, and use a YN-662-TX dedicated transmitter, so that you have all the radio advantages, as well as TTL and HSS. You can also do this with YN-586EX flashes, but you'd have to add on YN-622 units as radio receivers.
Your YN-560II can also use a a YN-622 trigger to be fired remotely, but as it's a manual-only flash, the situation is identical to the RF-602/603/605 scenario above: you still have to make all the settings on the flash itself.
Keep in mind, however, that Yongnuo is not the only game in town, merely one of the cheaper options. And that their triggering systems are more or less incompatible. Missing RF-603 and YN-622 triggering tends to involve stacking triggers. Other brands like PocketWizard, RadioPopper, and Phottix have better reputations and allow you to mix both manual and TTL triggers, which might be worth the higher unit costs, vs. having to entirely replace everything with Yongnuo models.