I am told that my tires' treads are "4/32" and that therefore the tires should be replaced "soon". What does 4/32 mean in this conrtext? What fraction means do it now?

The simple rule is take a penney and insert Lincoln's head into the tread over the width of the tire. When it reaches Lincoln's head the tires need to be replaced now.

Measurements are typically hard to do without a very thin ruler or caliper.

You can pick up a tread gauge at any auto parts store for $3. Poke it into the tread at three places--each edge and in the middle. New tires have between 8 and 11/32" of tread. Minimum tread depth varies depending on which state you live in. But anything 3/32 or less is really not safe. At 4/32, you're right on the edge.