A "golden birthday" is the birthday where your age is the same as the date of the month of your birthday. My birthday, for example, is on October 21, so my golden birthday happened when I hit 21, which was unfortunately too early for me to appreciate it, because I only found out about the concept a few months ago, at my grandfather's funeral, from a younger cousin of mine, who was complaining about how she is going to be the last one in her family to have a golden birthday. (A bit surreal, the whole thing, but so it goes, with life and death.)
At least my birthday does not fall on the first few days of the month, so that I would have only had the chance of being dimly aware of its passing, and is not on a Leap Year Day, so that I would not have my golden birthday until I hit the ripe old age of 116 (or 120, depending on what century it was.) It is interesting to note that if you were born on Leap Year Day, you cannot have a golden birthday in the traditional sense, because 29 years after your date of birth is not a multiple of 4, which means that it cannot be a leap year also. Sorry; that's the way the cookie crumbles.
Trying some Web searches, the only references to its origin were that Joan Bramsch, a "family person, educator, premium toy designer, writer and a publisher", claims to have come up with the idea as something special for her children, and that the question of what a golden birthday is was answered in the Chicago Tribune on July 9, 1978, which is the same year I was born, and Ourselves and Our Children was published, incidentally.