When you paint weights I guess you already know that you have to choose a bone from the list in the paint window so that Maya knows which bone you are weighting the skin to. Yes, white means you are painting weight. Black means there is no weight. If the skin is moving when you bend a joint it means that the skin is being influenced by more than 1 joint. If the skin is painted white yet it doesn't move then maybe something is turned off. I'm not sure what though.

here's the thing about weights; The weight is located at the vertices and has a value of1. 1 is completely white. 0 is completely black and all values inbetween 1 and 0 are a mix between black and white. If you have a spot on your model that stretches when you move a joint it means that the total value of the weight is split up between 2 or more joints. Maybe each joint has .333 for its weight value. You can look at the weight values by going into component mode. You can even edit them manually. The piant tool that you use to paint the weights is just an easy way to apply the weights. Instead of typing numbers, you paint white.

One more note to confuse you some more, the skin weight value MUST always equal 1 if it is to stay completely bound to the skeleton. So, for example, if the weight for all vertices is .5 then you moved the skeleton, the skin would follow, but not completely. Anyway, this is the point I'm trying to get to; if you adjust the weight up or down while you have a certain joint selected, the weight that you are painting is coming from another joint. If you are subtracting white from a joint that means that the weight value is being placed on another joint so that everything equals 1. You can put a HOLD on joints so that their value doesn't change but I'll leave that for you to figure out. Also, just because you don't see any white on a certain area of the skin, it doesn't mean that there are no weight values there. If a spot of skin has weight values like .01 then It might still look black even though there are weights there. This will cause the skin to move even though you can't see the weight influence. Maya has a cropping tool to delete really low weight values.

Weighting the skin can be frustrating but you shouldn't give up.