The "person" is Miss Liberty. You have what's called a Liberty Standing quarter. It's considered to be one of the most beautiful U.S. coins of the 20th century. The design was struck from 1916 to 1930. Unfortunately a defect in the original design caused the dates to wear off quickly, and the problem was not corrected until about 1925. As a result many SL quarters are dateless.
On the front of the quarter is George Washington's face. On the back is an eagle (there is no person depicted for the reverse side).
That's called a Standing Liberty quarter. They were minted 1916-1930.
Front: George Washington Back: Eagle
Not enough information. Is the date missing due to wear? If so, is the coin a Washington quarter or does it have an older design - Liberty standing on the front and a flying eagle on the back, or the head of Liberty on the front and a spread-eagle on the back? If it's a Washington quarter, look at the edge. If it shows copper then it's a modern quarter and is worth only 25 cents. If the edge is silver colored or it's one of the earlier designs you have a 90% silver coin that might sell for $2.50 or $3 for its metal content.
No, 1924 is a Standing Liberty quarter, Washington quarters started in 1932.
Two types of quarters were made in 1916. The most common type is called a "Barber" quarter after its designer Charles Barber. This design has Miss Liberty's head on the front and a spread-winged eagle on the back. The other design is called a Standing Liberty quarter and is much scarcer. Its design features a full-length image of Miss Liberty on the front and a flying eagle on the back. Any such coins should be inspected by an expert to ensure they're genuine. There's more information at the Related Questions.
Your coin is normally referred to as a "Barber" quarter after its designer, Charles Barber. There's more information at the Related Question link.
The 1914 US Barber quarter shows "Miss Liberty' on the front of the coin. Not a depiction of a real person.
The date is 1857 not 1837 It's a Flying Eagle one cent coin, that was only struck from 1856 to 1858
The US has never made a gold 25 cent coin.
There are Flying Eagle cents and there are Indian Head cents but they're not the same coin. Please look again and post a new, separate question with the coin's date/ As the names might suggest, Flying Eagle cents have a picture of a soaring eagle on the front while Indian Head cents have a stylized Native American image*. Also Flying Eagle cents were minted only in 1856-1858 but the first Indian cents were minted in 1859 so there was no overlap in production. (*) The image technically represents Miss Liberty and was modelled by the designer's daughter, but everyone calls them Indian Head cents anyway!
The front of the quarter had lady liberty with a cap and wreath on her head. She sort of had a Greek look, and it says "In God We Trust" on top, and the year the coin was made on the bottom. On the back, there is an eagle, or more specifically a heraldic eagle that has a banner in it's mouth stating "E Pluribus Unum", and on the top it says "The United States Of America", and the bottom says "Quarter Dollar", and there is thirteen stars above the eagle, yet under where it says "The United States Of America".