Example sentences
jīn今tiān天shéi谁shēng生rì日？
Whose birthday is it today?
shēng生rì日kuài快lè乐！
Happy birthday!
About this word
生 is "to be born" and 日 is "day", so 生日 is the day of birth. It is a noun, and the verb it needs is 过, so "to celebrate a birthday" is 过生日 and the standard greeting is 生日快乐. The word only ever means the anniversary. For the date you were actually born, the one that goes on forms, Chinese says 出生日期, "date of birth". 日 is the bookish word for "day" and mostly lives inside compounds now, while 天 is what people say in speech.
The story behind the character
生
A sprout poking out of the ground: the bottom line is the soil, the rest is the shoot coming up. Xu Shen wrote 生, 进也, 象草木生出土上: "to grow, like a plant breaking the surface." So the 生 in 学生 and 先生 started as a blade of grass sprouting.
日
日 is a drawing of the sun: a circle with a mark in the middle. In the oracle bones it is a round disc with a dot or short line, and the dot was likely there to keep it from being mistaken for a plain circle. Xu Shen wrote 日，實也, the sun is full and solid, never waning.