An anniversary marks the date on which an event happened — a five-year wedding anniversary occurs exactly five years after a wedding.
When someone refers to "our anniversary," they are typically talking about the day they were married. News organizations regularly recognize the anniversaries of important people's births and deaths, plus anniversaries of significant news events and national tragedies, like Sept. 11 2001 or Dec. 7, 1941. This word comes from Latin; remember that annus means year and you can relate the word to annual. Sometimes the world is used more loosely, like when new couples celebrate their one-month anniversary.