Learn Russian: Моя сестра замужем.

Possessive words like "my" must match the gender of the noun they belong to. Because сестра (sister) is a feminine noun, you must use the feminine form моя instead of the masculine мой.

Where is the verb "is" in this sentence?

Russian usually skips the verb "to be" in the present tense. Instead of saying "My sister is married," you simply put the subject and the state together: "My sister married" (Моя сестра замужем).

I've seen a different word for "married" before. Why are we using замужем here?

Russian has completely different words for "married" depending on the person's gender. замужем is used only for women (it literally comes from the phrase "behind a husband"). If you were talking about a man being married, you would use the word женат instead.