Ancient history

Is a coin with buffalo on one side and an Indian head the other dated 1877 real coin?

There is no coin called a "Buffalo Indian Head". The United States Mint did produce a coin called the Buffalo Nickel from 1913 to 1938, but it featured a Native American on one side and a bison (often erroneously called a buffalo) on the other side. Additionally, the date 1877 does not correspond to any known Buffalo Nickel issue. The first Buffalo Nickels were minted in 1913. Therefore, the coin you are referring to is not a genuine United States coin.