Serene is related to the Latin word serenus "peaceful, calm, clear".  Serene was originally used in English, as in Latin, to describe calm weather. By the mid 1600s, however, it was used figuratively to describe a calm, untroubled person as well.

The word "serene" first appeared in Middle English around the early 1500s, originating from the Latin word "serenus" which means "clear, cloudless, untroubled".  OED's earliest evidence for serene is from 1508, in the writing of William Dunbar, poet and courtier. 

Examples of serene in a sentence: 
He appeared calm and serene as he always did. 
Her eyes were closed and she looked very serene. 
There wasn't a cloud in the sky to obscure the deep serene. 
He didn't speak much, he just smiled with that serene smile of his.
We sailed serenely down the river.
She carried on serenely sipping her gin and tonic.

3 minutes ago, drpaladin said:

Serene can be used as a form of official address for some titled persons, as in His or Her Serene Highness or Majesty. This comes from one of its lesser meanings, august.

That's my birth month, so does that imply that I'm serene?  

Just the facts:

SERENE Definition & Meaning - Merriam-Webster