The answers you’ve gotten so far are somewhat scattered, so let me put together a recap.
The key thing to remember is that, although U.S. coins today are basically tokens, originally they were made out of valuable metals, and a coin was a way for the government to certify that it was composed of metals worth the face value of the coin. (Contrary to popular belief, you are allowed to melt down coins if you want to).
Dimes (10 cents), quarter’s (25 cents), half dollars (50 cents) and dollars were generally made out of silver–a silver dollar weighed 10 times as much as a silver dime, and so forth. This meant that a dime came out pretty small, although for a while they even had a 5-cent silver coin, called a “half-dime” which, being half the weigh of a dime, was a somewhat pitiful thing. A silver dollar was pretty hefty, so larger denomination coins were made out of gold.
For the smaller valued coins, less valuable metals were used (for example zinc, copper and nickel) which is why the 1 cent (penny) and 5 cent (nickel) coins are bigger than a dime, since they were made out of cheaper materials.
As other pointed out, in the 1960s the government removed the silver coins from circulation, and replaced them with dimes, quarters, half-dollars, and dollars made of non-valuable metals. But they kept their same size, to make the transition easier. (Later, they also changed the composition of pennies to coated aluminum.)