Although the first Chick-fil-A restaurant opened in 1967, the name was originally used for a chicken sandwich at the Dwarf Grill in the Atlanta suburb of Hapeville, GA. The name was first spelled as "Chick-Fill-A". 
The name was changed to the current spelling in 1963.
The logo was redesigned in 1964, resembling in appearance what is used today. This logo would be applied to the first Chick-fil-A restaurant in 1967.
In 1975, the logo was modified to have thicker lines.
Same logo as the second one, but darker in color. When used on outdoor signage, everything in the logo was red. This still remained on some markets until 2013.
Same logo as before, but this time the all-red coloring is more widely used, and the handwriting is now thinner. The new handwriting was used in tandem with the previous thicker handwriting. This logo is still used at some locations.
Chick-fil-A made a refresh to their logo by making the color brighter, the C smaller, and closing the chicken's beak.