The United States minted a variety of different quarters over the years, so the weight can vary based on the quarter. However the modern clad quarters weigh 5.670 grams or 0.2 ounces. Most of the silver quarters minted before 1965 weigh 6.25 grams or 0.22 ounces.
Clad Quarters
Any United States quarter minted after 1964 is a clad coin that weighs 5.670 grams. Essentially these are your everyday quarters that will be found in your change. They consist of 91.67% copper and 8.33% nickel.
Silver Quarters
The more-common silver quarters weigh 6.25 grams. Every silver quarter minted from 1873 to 1964 has the same weight of 6.25 grams. This includes all of the silver Washington quarters, Standing Liberty quarters, and Barber quarters. These quarters consist of 90% silver and 10% copper. You are unlikely to find them in circulation but you can buy them as bullion or individually as collectible coins.
Early Silver Quarters
The rare, early-issue silver quarters have different weight variations, depending on the series. Note that these older quarters are completely out of circulation and can only be bought as collectible coins. So unless you are a collector then you will likely never come across them.
Seated Liberty Quarter
The Seated Liberty Quarter was first minted in 1838 with a weight of 6.68 grams. The weight changed to 6.22 grams in 1853, and then in 1873 there was a small increase to 6.25 grams.
Capped Bust Quarter
The Capped Bust quarter was first minted in 1815 with a weight of 6.74 grams. For the years 1837 and 1838 the weight changed slightly to 6.7 grams.
Draped Bust Quarter
The Draped Bust quarters were minted from 1796 to 1807. These valuable quarters each weigh 6.74 grams.
See also:
How Much Does a Penny Weigh?
How Much Does a Nickel Weigh?
How Much Does a Dime Weigh?