What is a birthday cake?

A name-day cake is the same a birthday cake. There is no specific recipe for a name-day cake.

In Russia, historically, people celebrated their name day instead of their birthday. A name day is the day in the Church calendar dedicated to the saint, after whom one is named. So, if I my name Ivan, my name day is April 27, regardless of when I was born.

With more first names of foreign origins, as well as made-up names, making their way into the Russian culture in the early 20th century, it became impractical to celebrate name day, as many kids would not have a corresponding saint. Still, the birthday is often colloquially referred to as one's "name day" (imeniny in Russian) and the birthday cake (which would sound very awkward in Russian) is called a "name-day cake" instead.