What I understand is that a Dudley can refer to either a chocolate-pigmented yellow or a non-pigmented yellow. Basically, a yellow lab without the dark pigment around eyerims, etc. "Snow nose" in an otherwise dark-pigmented yellow doesn't count.
Yes Julie is right. Basically a Dudley is a yellow coated dog with chocolate skin. Yellows that have black noses, lips, and eyerims carry at least one black gene (those black noses fade and again as Julie mentioned this is normal for yellows and is considered a "snow nose" and not a Dudley). Those yellows that carry two chocolate genes have brown/chocolate noses, lips and eyerims.
Dudleys can be very light in color and have a corresponding pinkish/brown pigment whereas others are darker in color all over.
They can be produced in litters in which both parents either are or carry BOTH yellow and chocolate. This can happen when breeding many different combinations like black to black, yellow to yellow, and black to yellow. They are often misunderstood and people think it's a simple mix of yellow and chocolate so it's often thought of as a "no no" to breed a yellow to a chocolate when in fact most of the time all pups produced by that combination will be black. It's more complicated as you need to know what the dogs carry genetically.
Here are two photos of Dudley yellows. They are also sometimes called "liver nosed yellows" or "NBP" = no black pigment.