What people mean when they talk about the weight of a sheet of paper or card is effectively the thickness of that piece of paper. There are two conventions currently in widespread use for paper weights.
The first, as specified by ISO 536 is used in Europe and most parts of the world that use ISO paper sizes, apart from the few countries that still use US paper sizes, is grams per square metre (g/m2 or commonly gsm). This commonly called grammage in non English speaking countries, a term we will use on this site to differentiate it from the North American basis weights.
The second is used in those countries that use US paper sizes such as the United States and Canada. This is known as the basis weight and is usually specified in pounds (lbs). The basis weight is the weight of a 500 sheet ream of uncut paper (see Reams & Quires - Paper Quantities Explained for more information on the ream).
Ream of 80gsm office paper - showing clearly g/m2 on the packaging.
The weight of paper in countries that use ISO paper sizes is defined by ISO 536 Paper and Board - Determination of Grammage. This standard defines grammage as grammes per square metre (g/m2) thus the weight of a single sheet of A0 paper defines the grammage of that paper type as the A0 size is defined by ISO 216 as having an area of 1 square metre. An A0 sheet of 80gsm paper will weigh 80 grams, an A0 sheet of 100gsm paper will weigh 100 grams and so on.
Note gsm is commonly used rather than g/m2 in most day to day use. This came about because of the inability of early computer packages to display superscripted characters - especially stock control, accounting and invoicing systems.
Paper that is used in offices is usually between 70gsm and 80gsm, with 80gsm being the most commonly found weight. Some accountants and solicitors use heavier weight paper ranging from 90gsm to 120gsm for formal correspondance. Above 120gsm come various thicknesses of card with 160gsm and 200gsm being most commonly used for file dividers. Newspaper ranges between 45gsm and 50gsm.
See this page for the weights of individual sheets of 4A0, 2A0, A0, A1, A2, A3, A4, A5, A6, A7, A8, A9 and A10 for various commonly used grammages.
Ream of 65lb Cover Stock.
Basis weight is a much older convention for defining the weight of paper and was widely used in the UK and Europe as well as in North America, where it is still used, up until the international standardisation of paper sizes by ISO 216 and its national predecessors in European countries.
Basis weight is defined as the weight of a ream of the uncut paper in pounds (lb) (Note: the # is often used in place of lb when referring to basis weights). The uncut paper sizes being the size used in the manufacturing process before the paper was cut to size. There are many different types of uncut paper, those in common use for the production of office paper and card are Bond Stock, Cover Stock and Index Stock. The word Stock is often dropped when referring to the different types of paper in their finished cut sizes.
Thus if you have a sheet of 20lb Bond Letter paper it will be the exactly the same thickness as 20lb Bond Legal, although the two sheets will weigh different amounts because of the difference in size. However a sheet of 28lb Bond Letter size paper would not be the same as a sheet of 28lb Cover Letter size paper as the uncut sizes for Bond and Cover aare different.
Paper that is used in offices is most commonly 20lb or 24lb Bond. Often suppliers will omit the words Bond and just give the basis weight as 20lb or 24lb, although Index and Cover are almost invariably used as this differentiates the other types from the 'default' Bond. 60lb and 65lb Cover and 90lb and 110lb Index are types of light to medium card often used as file dividers, with 80lb, 90lb and 100lb Cover being heavier weight card.
Newspaper has its own stock type, Newsprint, that measures 24" x 36" in uncut size. This is much larger than the Bond and Cover uncut sizes, thus the most common 30lb Newsprint is actually a much thinner paper than 20lb bond. The following page covers the different stock types generally available.