The Atlanta Braves inherited their identity from Milwaukee (who previously inherited theirs from Boston), when the Milwaukee Braves moved to Atlanta for the 1966 season they used the exact same logo, uniforms, and colour scheme. All that changed was the block 'M' was swapped out for the 'A' which the Braves still wear today.
The Braves nickname debuted in Boston, half a century before they ended up in Atlanta. After a slew of other team names including Beaneaters, Dovers, and Rustlers, the 'Braves' nickname was finally introduced in 1912. Team owner James Gaffney was a member of the Tammany Hall NYC political organization which used a Native American chief as its symbol. After an ownership change in the 1930s the club was briefly renamed the Boston Bees but switched back to Braves after just five seasons.
During their time in Atlanta, the Braves have had two distinct eras of branding, the present era began in 1987 and borrows heavily from their original uniforms worn in 1966 and 1967. During the 1970s and most of the 1980s the Braves went with a more modern look (for the times) switching to a blue and white pullover with a lowercase 'a' on their cap. It was in this uniform set that Braves' slugger Hank Aaron broke Babe Ruth's all-time home run record in 1974.