Getting more specific, people identify papers in terms of basis weights. Since coated papers are more compressed (calendered), you can't necessarily keep the same weight when you switch grades. For instance you may go from 60# uncoated to 70# coated sheet to keep the same thickness. That's why papers are usually referred to by weight and grade.

What is basis weight exactly? It is the weight of 500 sheets of paper cut to a standard size. So 500 sheets of 25"x38", 60# offset weighs 60lbs. The standard size for bon is 17"x22"; for text, offset and coated 25"x38", and for cover 20"x26". That's why two similar sheets of different grades may have very different basis weights, for instance, 24# bond and 50# offset.

Different grades of paper cover different ranges of weight.

TEXT -- Ranges in weight from 60# to 100#. But the most common weights are 70# 80#.
OFFSET -- Usually a 50# to 70# stock.
COATED BOOK -- Generally goes from 30# to 70# for web, 60# to 110# for sheetfed.
COVER -- Comes in 60#, 65#, 80#, 100#, 110#, 120# and 130# weights.