The 25c denomination is historically derived from the usage of the Spanish colonial 8 reale or "Spanish Dollar" coin in British colonial America. The Spanish Dollar (and smaller denominations such as the 4 reale, 2 reale or quarter dollar, 1 reale, and half reale) is estimated by some to have comprised half the coinage in colonial America so it's importance cannot be understated. As well as being minted in smaller denominations the the 8 reale coin was cut into halves, quarters, and eights to provide smaller change. The tradition of the "quarter" continued in 1796 when the first was minted for the US and has continued to date. Interestingly the USA toyed with a 20 cent coin from 1875 to 1878 but it failed to take off. A further point of interest is that a reale was known colloquially as a "bit" and a quarter "two bits", a term the quarter is sometimes known by.

As to the denominations of the notes, the 1, 2, 5, 10, 20 and 50 notes follow the 1-2-5-10-20-50 number series. The US does have $2 note but it is not generally used. This number series uses prime factors of 10 (1x10, 2x5, 5x2, 10x1, 10x2, 10x5) which makes arithmetic easier.

A lot of other countries use this same numbering system for both their coins and notes, for example Great Britain, Australia, New Zealand, and the Euro circulating countries. Countries such as Canada and Panama still have a 25c coin but use the 1-2-5 numbering system for their bank notes.

References

Jordan, Lewis - Colonial Coins - Section Contents. 2016. Colonial Coins - Section Contents. [ONLINE] Available at: http://www.coins.nd.edu/ColCoin/ColCoinContents/Contents02.html. [Accessed 24 January 2016]

Colin Bruce, 2007. Standard Catalog of World Coins 1701-1800. 4 Edition. Krause Publications

Colin Bruce, 2010. 2011 Standard Catalog of World Coins 1901-2000. Thirty-eighth Edition Edition. Krause Publications