Here's my understanding of paper weight, from a novice's point of view:
BASIS WEIGHT: The basis weight of paper is the designated fixed weight of 500 sheets, measured in pounds, in that paper's basic sheet size. I'm not going to explain basis weight any more than that- too complicated. You can google more if you want to muddle up your brain. Suffice it to say, the basis weight is the weight number you read on the label when buying paper, such as "20lb bond."
PAPER TYPE: Judging by the basis weight is not enough, however. That number is only relative to the type of paper being used. The two general paper types are: Text Weight and Cover Weight. The various Text weight papers are those you mostly associate with writing paper, copy paper, ledger paper and such - or the interior of books. Text weight paper comes as Bond/ Writing (sometimes called Text) and Book Text. You will find the Bond/Writing Text at most office stores. Copy paper is in this category.
Cover Weight types are stiffer papers you mostly associate with the outside of books or reports. Bristol, Index, Cover Stock, and Card Stock papers fall into this group.
The reason it is important to identify the type of paper as well as the basis weight is that the numbers of the two can sometimes be the same. Here the confusion begins!
GRAMMAGE: The grammage (how many grams a paper weighs) is a simpler indication of weight, from a consumer's point of view. However, paper in the store is rarely identified this way. (For example 20lb bond/text has the same grammage as 50 lb book weight. 100lb text has the same grammage as 67lb cover. 32lb bond and 80lb book are the same weight.) Are you confused yet? The website "Paper-Papers" has the simplest, best chart for paper weight comparisons I've found. See it HERE. I've copied this chart and now keep it tacked to my paper storage closet.
BRIGHTNESS: In judging white colored paper, it is important to look at the brightness number (not always listed, unfortunately.) On their own most whites look about the same, but next to each other the differences show up. Knowing the brightness can be useful when you are comparing papers or switching from one brand to another. Generally, recycled papers tend to be less bright because they are unbleached or treated differently.
HELPFUL HINT: I find it helpful to keep a paper sample file for myself. Take a ring binder (I like to use a personal planner size so it's easy to carry to the paper distributor,) divide it into text and cover sections, and keep in it a page of each type of paper you use or want. On it write the company brand, the name of the paper,color name, the paper weight, grammage , brightness, and any notes you want, such as where you bought it or how you've used it successfully. For example:
Neenah Classic Linen Cover Stock, Solar white, 80lb basis/ 216 gm, Xpedx --Use: wedding keepsake card
--------------------------------------------------------------All of these papers are used for printing and copying purposes. None of this has any relation to Art Paper. That is another discussion.