Is it possible to trigger Nikon SB-600 with.. say, any flash, wirelessly? If not, what speedlights are capable of this?
What's this feature called, so I can search for more details?
In Nikon speedlight terminology, the feature you want is SU-4 mode. The SB-600 does not have it, the SB-700, -800, -900, and 910 do. There are similar "dumb" slave modes in the SB-26 and SB-28DX.
The terminology is confusing, because of the two separate types of optical slave modes: "dumb" for see-flash-fire-flash tripping, or "smart" CLS full-TTL-bells'n'whistles tripping. You do sometimes have to take extra time looking over the specs or doing additional research and googling to determine which type of system is being described.
The only optical slave in the SB-600 is a "smart" one for CLS communication. CLS slave communication means that you have more control over the remote flash, as well as iTTL, FP mode flash (HSS), and rear-curtain sync. "Dumb" slaves are a manual-only method of triggering, all you can do is tell the remote flash to fire--you'll have to set the power on the flash itself, and features like FP and iTTL are not possible.
You can purchase add-on "dumb" optical slaves (e.g., Wein peanut, Sonia optical slaves), but they typically need some way to communicate with the flash, and the SB-600 does not have a PC sync port, so your best chance is to find one that attaches to the hotshoe.
There are a ton of 3rd-party flashes (Metz, Nissin, Yongnuo, etc. etc.), both TTL-capable, and manual-only, that exist that have built-in "dumb" optical slave modes, as well as flashes with CLS slave capability. For example, the Yongnuo YN-568EX can be used as a dumb slave in modes S1 (triggers on first flash burst) and S2 (triggers on second flash-burst, so it can avoid early tripping on a TTL pre-flash for metering). And can act as a CLS slave in Sn mode (and a wireless eTTL slave in Sc).
The SB-600 does not offer a 'dumb' optical slave trigger mode, only CLS/AWLS. So you'd be better off getting another flash that uses CLS unless you have reason not to. Don't forget that most Nikon bodies will act as a CLS master via the built in flash though.
You'd require a separate trigger as mentioned by dpollitt's answer.