Vinetting is the occurance of a dark or black areas showing up in the pic due to the step down rings, filters, hoods or maybe the lens itself. Its due to the nature of the pics being rectangular shaped and lenses being round. Thats why those lenses with very wide angle will come with those petal hoods which is designed to prevent vignetting. If the hood is not petal shaped, the hood would be captured as well and shown as dark corners on the pic. This effect is also vignetting.
People step up or down to save cost, not all lens can with the same diameter. But normally people step up than down to prevent vignetting.... Wide andle len will be worse cos of the wider field of view.
With a step-up ring, you can use one set of filters on two or three different thread-sized lenses. I have a whole set of 62mm filters that I use on my 62mm and 52mm sized lenses with the aid of the 52mm-62mm step-up ring. The attached image gives you an example of of its use - a 62mm B12 filter coupled to the 52mm thread-sized 55mm f/2.8 AIS lens via the 52mm-62mm step-up ring.