I'm answering this under lenses because the discussion often devolves to a lens issue.
To be clear, "photographic equivalence" means being able to take the exact same picture with different equipment. Now, it should be obvious that if I take a picture with a D7200 and a 35mm DX lens set to f/4, put that camera down and picked up another D7200 with a 35mm DX lens set to f/4, I should get the same exact image (assuming all my camera settings were the same, of course).
When Nikon introduced the DX DSLR (and Olympus the 4/3 DSLR) something different popped onto the scene from the 35mm film days. Their capture areas are different in size and lenses don't seem to work the way they used to. Let's leave out sensor differences for the moment and just concentrate on the format/lens relationship. (For a discussion that starts with sensor, see this article on sansmirror.com.)
Let's assume that we're comparing a DX DSLR (1.5x crop) to an FX DSLR. To take an equivalent image five things need to be the same: perspective, size of output, shutter speed, angle of view, and depth of field. The first three are very easy to deal with.
Perspective stays the same if we shoot in the same place, so let's do so ;~).
Size of output is easy: we'll just make 8x10" prints from both the DX and FX image we shoot.
Shutter speed is a snap, we'll just...oh oh, we might have a little bit of a problem with that if the aperture changes, but let's just go ahead and say we have a stationary subject and the shutter speed will be fast enough no matter what to cancel out any camera motion.
Angle of view has three variables: focal length, imaging circle, and size of capture area. I'm going to assume that we're using a lens with a big enough imaging circle. If we put that 24mm lens on a DX camera and on an FX camera, the angle of view that is seen is different due to the size of the capture area: DX crops. On DX we get 52° horizontal, on FX we get 74°. Oops, that's not "equivalent." So we'll put a 24mm lens on DX (52°) and a 36mm lens on FX (52°). We needed almost exactly 1.5x the focal length for FX that we did for DX to get to equivalence (technically it varies slightly with different DX sensors, but not enough to worry about).
Some people believe that depth of field stays the same for the same perspective (the Merklinger thesis), but even if we were to use Zeiss we'd find that we don't have to worry about this variable (it just takes a lot of math to get there, which I'm not going to do here). We're using different focal lengths, so guess what, it turns out that we'll need a different aperture to get the equivalent photo, simple as that.
Which brings us to the grand conclusion that usually precipitates the asking of this question: you'll need to use about one stop faster aperture on DX to get the equivalent image as FX. But you can get the equivalent image.
Likewise, m4/3 is about a stop off from DX, so to get the look of a 50mm f/2 lens (FX) on DX you'd need a ~35mm f/1.4 lens (one stop difference), and on m4/3 you'd need a 25mm f/1 lens (two stops difference). Which explains why I have a 25mm f/0.95 lens for my GH4.
Unfortunately, you can't always get exactly the right lens for all needed equivalencies: for example, we can use a 200mm f/2 lens on our DX body to get what we got from our 300mm f/2.8 lens on our film or FX bodies. Nikon makes both. But if we wanted to match 200mm f/2 FX, we'd need a 135mm f/1.4, which doesn't exist. Of course, if we were shooting that 200mm FX at f/8, we could use the 135mm f/2DC set to f/5.6 to get basically the same shot. All of which will probably lead you to another question ("but is the light hitting a DX sensor the same as that hitting the FX sensor?) which is another question for another day, and isn't a lens question anyway.