Hi Joe.G, Rhode Island Reds that have been bred to standard have a deep mahogany color. Their bodies have the characteristic "brick" shape as outlined in the SOP and oftentimes the tail feathers are black with a green sheen. Production Red can mean one of two things- a red-colored chicken derived of RIR stock but does not meet the standard or it can be a red sex-link female. A New Hampshire Red is a separate breed altogether. While they are also red, they have more of an orange-y red look to them. There are threads devoted to RIR and New Hampshires on this site. All three birds lay brown eggs.
Since I don't keep either (I do have a few production-quality red hens), that's the best explanation I can give you.