The old film format measurements still exist because of the transition from film to digital - it's hard to let go of what has been used commonly for so long. So '35mm equivalent' still refers to 35mm film cameras. Digital cameras have sensors that come in so many sizes, and the size of the sensor affects how much of the lens' frame actually falls onto the sensor...a smaller sensor is not using the full lens opening circle, which is what is commonly referred to as the 'crop factor' - in other words, how much would you have to crop down the 35mm frame to equal the capture of the smaller sensor. Even P&S cameras have crop factors...but you never really hear them because they usually give you the camera's focal range in '35mm equivalent' terms. But you can always find out what the true lens focal range is by looking on the front of the lens itself...even on P&S cameras. The crop factor is the amount you have to multiply the lens' focal range in order to equal the 35mm sensor/film.
Full frame DSLRs are the ones that match the 35mm film frame in size - so those don't have any crop factor. APS-C DSLRs have smaller sensors than full-frame, and have a crop factor of 1.5x (Nikon, Sony, Pentax) to 1.6x (Canon). Olympus and Panasonic's 4:3 format has a crop factor of 2x. P&S cameras can range from 5x to 7x crop factors.
So, if you put a 50mm lens on a full frame DSLR, you get 50mm. If you put it on an APS-C DSLR, you get a crop equivalent to 75mm (or if using a Canon, you get 80mm). If you put it on a 4:3 Olympus, you get 100mm. If you were to design a P&S camera with a 1/1.7" sensor and a lens with a true focal length of 50mm, you'd get the equivalent of 300mm! Big crop factors on P&S cameras...which is why you'll notice the actual lens on most P&S cameras have a really small lens, with a true focal length range of something like 6mm to 72mm (which would be on a superzoom camera, delivering the equivalent of 36mm to 432mm).
If you have an APS-C camera, you want to look for a lens that will give you the cropped equivalent of 50mm - so rather than look for a 50mm lens (which will be more like 75mm), you would want to look for a 35mm lens, which would be close to the standard '50mm' focal length with the crop factor considered. If you have a full frame camera, then you want to look for a 50mm lens, as you have no crop factor. If you're shooting with a 4:3 sensor camera, then you'll want to look for a 25mm lens.
Sony DSLR-A68 / Sony 18-250mm / Minolta 50mm F1.7 / Tamron 150-600mm / Minolta 300mm F4 APO
Sony A6300 / 18-55mm F3.5-5.6 / 55-210mm F4-6.3 / 10-18mm F4 / 35mm F1.8 / 16mm F2.8 / FE70-200mm F4 G OSS / FE70-300mm F4.5-5.6 G OSS / via manual adapter, lots of Pentax K mount, Konica K/AR mount, and Leica M mount manual lenses