Within the fantasy world of Game of Thrones, a "nameday" is the equivalent of a "birthday" in the culture of Westeros.
In the Seven Kingdoms, babies are named the same day they are born. Therefore, a "nameday" is functionally the same thing as a "birthday", as both terms indicate the day a person was born on. The only slight difference is that in Westeros, the act of naming is what is counted, instead of the birth itself. People receive presents from friends and family on their nameday.
In the Books
In the A Song of Ice and Fire novels, people count their age by each annual "nameday", but this is functionally equivalent to a real-life "birthday" because babies in the Seven Kingdoms are named the same day they are born, hence the phrase "naked as my nameday", and a character says that war has made the Kingsroad dangerous to travel, when once a girl could walk down it in her "nameday suit" and be safe. That is, it is not like the real-life Jewish brit mila circumcision ceremony during which male babies receive their Hebrew name, which takes place eight days after birth. A "nameday" is equivalent to a "birthday", it's just a quirk of the characters' culture.
The only exception to this on the continent is north of the Wall, where the wildings live in such a harsh and unforgiving environment that infant mortality is quite high. Therefore, the wildings only name their children when they survive to two years old. While Gilly named her baby "Sam" at the end of Season 3, in the books her child explicitly has no name because he isn't two years old yet (less than two years pass between her child's birth and the end of the current and fifth novel, A Dance With Dragons).
Real-life medieval societies, upon which Westeros is based, did not possess a concept of "adolescence" as modern-day societies do. There was no concept of an intervening live-stage between childhood and adulthood. Thus as soon as a girl flowered (menstruated), she was considered a "woman", and could be married.
In contrast, the legal age of majority for boys in the Seven Kingdoms is considered to be sixteen years of age. Again, there is no concept of being a "teenager" or "adolescent". A boy who has turned sixteen suddenly becomes "a man grown".
This may have been changed in the TV series, in which all characters have been slightly aged-up (usually about two years) due to censorship issues. In "Cripples, Bastards and Broken Things", Samwell Tarly says that his father forced him to leave home and join the Night's Watch on his eighteenth nameday as he was "almost a man now", not his fifteenth nameday as in the books.
Bronn does use the term "birthday" instead of "nameday" in "The Ghost of Harrenhal", when referring to what Joffrey did to his "birthday present"; as does Robb Stark in "Baelor"; however these may have just been ad libs by the actors, and thus dialogue errors.