As an example, a 35mm lens on a crop body (such as Nikon DX) is said to be "equivalent" to a 50mm lens on a full frame body. Equivalent here means that they will have the same field of view, but what other aspects will change or stay the same between the two? I'm thinking about things like perspective, bokeh, and depth of field (assuming the same aperture being used on both lenses).
Assuming the same scene and focal lengths chosen to match field of view:
Perspective will be the same, as it is a function of subject distance.
Exposure will be the same for he same f-stop.
Depth of field will be different for the same f-stop, as it is a function of absolute aperture size, 50/2.8 is a bigger quantity than 35/2.8, so the 50mm lens will have shallower depth of field.
Bokeh in the strictest sense is dependant on the lens design and construction so there is no overall trend.
The larger format will in general resolve more detail.
Calling it "equivalent" is just shorthand, and often not very accurate.
First, the lens doesn't know what body its mounted on. You can mount the same 35mm lens on a crop frame APS-C, a full frame, and on a view camera.
Calling the camera a crop-frame is the important one, it takes the same photo, but crops it. When you crop an image, you change it. If you shoot at the same camera/lens to subject length with crop and FF bodies, the perspective will be identical -- it will be easier to see when you crop the FF image.
Most definitely, the depth of field will not be the same. You get smaller depth-of-field when the sensor/film is bigger relative to the focal length of the lens.
It is well known that in order to get the equivalent focal length, you have to multiply the real focal length by the crop factor. Thus, a 35 mm lens on a DX body is equivalent to a 53 mm on FX (the DX crop factor is 1.52).
What is less known is that the same crop factor applies to the F-number. Thus, a 35 mm f/1.8 on DX is equivalent to a 53 mm f/2.7 on FX. By “equivalent” F-number I mean that in both situations you will get:
- the same light gathering power, i.e. the same number of photons collected from a given scene with a given exposure time,
- the same depth of field (c.f. this answer by mattdm)
- the same ability to blur the background, which is related to but not the same as depth of field.
Now, if you use the same exposure time, the two situations will obviously not give the same ISO. If you have ISO 200 at f/1.8 on the DX body, you would need ISO 460 at f/2.7 on FX. But you should nevertheless expect the same amount of noise, assuming both sensors have the same intrinsic quality.