Dimes, quarters, and halves dated 1964 and earlier, and dollars dated 1935 and earlier, were struck in an alloy of 90% silver. The government controlled the price of silver at that time and fixed it at $1.29/oz so the amount in each coin corresponded to its face value.
When demand for silver increased the government was forced to let the price float, and coins became worth much more for their metal content than their face value. In 1965 dimes and quarters were changed to cupro-nickel clad composition and the amount of silver in half dollars was reduced to 40%. By 1971 halves were changed to cupronickel as well, and the new dollar coins introduced that year were also made of the same metals. 1964-dated silver coins are not rare as collectibles. Most are worth their melt value only. That varies depending on the price of silver but has averaged 8 to 10 times face value during 2008 and early 2009.
Note that nickels and cents from 1964 DO NOT contain any silver, despite urban legends to the contrary.
(02/2009 - prior answer replaced because it was completely incorrect regarding both the percentage of silver in and metallic composition of U.S. coins)