The very lowest point of a steam locomotive's boiler - at the bottom of the firebox. The mud ring is, as the name suggests, a squared-off ring that joins the inner and outer sheets of the firebox at the bottom. The 'mud' part of the name is simply because mud collects there - more accurately, scale, impurities, debris (including, yes, mud, if the water isn't clean) and the like. These build up at the bottom of the water legs of the firebox until they are removed, either when the boiler is drained and given a thorough washout, or (partially) by the use of blowdown valves - opening valves at the bottom of the water space just above the mud ring while the boiler is in steam, blowing out hot water from that area and hopefully taking a good deal of the mud and nasty stuff with it.
Because of the collection of impurities and dirt, the mud ring is often the first part of the boiler to wear out and need replacing.