Brightness is reflected light. With lower brightness, overall contrast is reduced, and highlights are dulled. Brighter papers cost more, in general, since brightness is a result of adding costly additives like titanium dioxide to the stock.
Not all papers within a given grade category are equal, however. Differences in ink holdout, smoothness, opacity, the amount of coating, side to side consistency, and runability must be taken into account as well.
Papers come in a variety of surface types, and once again, individual stocks vary within a classification. The most common coated surfaces are cast-coated, gloss, dull, matte, silk and embossed; uncoated grades come in a wide variety of finishes such as smooth, linen, vellum, and felt. Each of these surfaces will provide different print quality and overall appearance. Each has its strengths and appropriateness for a particular job.
Most grades come in a variety of weights for both cover and text. Having a dummy made before you specify weights is invaluable, since it allows you to check for opacity (put a page of type behind one of the unprinted pages) and for the overall 'hand' of the piece.
Holdout refers to a paper's ability to hold ink on the surface consistently, so that it will dry in a sharper, more clearly defined dot and produce higher ink gloss. When ink is absorbed into the sheet, it spreads, creating a phenomenon referred to as 'dot gain'.
Printers 'fingerprint' their presses with a variety of different papers. In fact, they often have specific performance data for the combination of paper, press, and pre-press techniques being used. Printers can be invaluable in helping specify a sheet.
There are two types of opacity: 'apparent opacity' refers to the actual opacity of the unprinted paper itself; 'printed opacity' is affected by holdout, in that the lessened opacity is actually caused by absorption of ink. As ink is absorbed into a sheet of paper the printed opacity of the page decreases, causing the image to show through on the back.