Cakes were very different from what we know today, in ancient Greece they started off as bread-like, sweetened with honey. 

In the middle-ages, Germany used a sweetened bread-dough in the shape of baby Jesus to commemorate his birthday. It was later that the Birthday Cake truly emerged as a celebration item for young children, or kinderfest. 

At around the same time, medieval European bakers started to bake fruitcakes and gingerbread, they could last for months (and be re-gifted to others around Christmas, just kidding). In medieval England, objects such as coins and thimbles were mixed into the batter, the person who got the coin would be wealthy while it was said that the thimble finder would never marry. 

It was sometime in the mid-17th century that precursors of modern cakes (round ones with icing) first started to appear as refined sugar and moulds started to be accessible. The first icing was a boiled composition of fine sugar, egg whites and flavourings  This glossy, ice-like covering was poured on the cake and returned to the oven to form a hard covering. At this point cakes were still only available to the very wealthy.

In the 19th century, because of the technological advancements and mass production of the industrial revolution, baking ingredients became more affordable. Cakes began to be made with extra refined white flour and modern leavening agents such as baking soda and baking powder were invented. Butter-cream frosting began replacing traditional icings in the beginning of the 20th century. 

The first evidence of advanced baking skills was found in ancient Egypt but it was in ancient Greece that round honey cakes were first made to honour Artemis, the goddess of the moon. Candles were lit on the cake to make it glow like the moon.

In contemporary Western cultures, the birthday song is sang around the birthday person, while he or she stands before the cake ready to blow out the candles at the end of the song. The "Happy Birthday to You" song has overtime been translated in many languages. This song originates from the "Good Morning to All" song written by Patty and Mildred Hill in 1893. Little of us know that this song is actually copyrighted by Warner/Chappell Music and one cannot sing this song for profit without paying royalties, this includes use in film, television, radio, anywhere open to the public, or even among a group where a substantial number of those in attendance are not family or friends of whoever is performing the song.

In French-Canada, another song is sang, we sing the "Mon chèr ami, c'est à ton tour" song (in the link, it is not actually sang, but I couldn't find another version). If your culture sings a different song, I'd love to hear it. 

In ancient times, people prayed to their gods over the flames of an open fire. They believed that the smoke carried their thoughts up to the gods. Today, the birthday person makes a private wish before blowing out the candle(s). If you succeed in blowing it out in a single breath it is said that your wish will come true. 

While I couldn't find anything on this, it is a French-Canadian tradition (maybe it's like this elsewhere too?) that the birthday person cuts and serves a piece of cake to everyone before serving him or herself a piece. It is seen as form of respect and hospitality to the attendees.