The wedding is the official ceremony/party. (The party is usually called the "reception"):
I'm going to my cousin's wedding on October 7.
We want to have a band at our wedding reception.
The wedding will be at the church, and the reception will be at a restaurant. The marriage is the relationship in general, or the institution in society:
My parents have a strong marriage. They've been together for 35 years.
New York has just legalized gay marriage.
Over 40% of marriages end in divorce.
The word married describes the status of a person. Use married to (not "married with") when you describe the person's husband/wife:
My sister isn't married. She's single.
I've been married for 5 years.
He's married to a woman from another country.
She's married to a successful businessman.
We use marry or get married for the action of formalizing the marriage:
He married a woman he'd known since high school. They finally got married after dating for ten years.
She's going to marry a famous actor. They're getting married next month.