I often hear about sensors being full frame (which I think means 35mm sized) and others being cropped (if they are less than that) such as APS-C sized sensors.
My question is what is so special about the size 35mm? Why is it used as the standard against which everything else is measured? Is it the largest sensor possible or does it have some other characteristic that makes it stand out?
For digital cameras, it's purely due to historical reasons - 35mm was the dominant size for film cameras and cinematography. As for why film cameras ended up with 35mm, I'd suggest 35 mm film and 135 film on Wikipedia as a good place to start. It's also worth noting that "35mm" is not actually the size of the image, which is 36x24 mm, but the width of the film.
It's certainly not the largest sensor available - a number of companies (Hasselblad, Phase One, Pentax) make "medium format" sensors which are larger than 35mm sensors.
Because the standard size for film cameras for a long time was 135 film which measures 35mm in width including the perforations and leaves enough space for a 36x24mm negative size. Since the Field of View (FoV) for any particular focal length lens is determined by the size of the film or sensor onto which the image circle is being projected, over time various focal length lenses became associated in photographers' minds with a corresponding FoV for that focal length when used with a 35mm film camera. By relating digital sensors of other sizes as a linear ratio to the size of a 36x24mm 'full frame' sensor, it makes the task of figuring the equivalent Angle of View (AoV) for any particular focal length lens a simple exercise of multiplying the the smaller sensor's crop factor by the focal length of the lens.