A guest to a bar mitzvah should dress up a bit, as if going to a wedding. Men should wear suits, and women should wear a dress or dressy pantsuit. Men are required to cover their heads while in synogogue, but most synogogues provide kippot, or yarmulkes, for the purpose.
Dress for a bar mitzvah may vary slightly depending on the style of the community and the lavishness of the reception thrown after the religious ceremony. Check with your host family to see if black tie is required. The friends of the 13-year-old celebrating his bar mitzvah aren't always held to the same dress code, and sometimes themed dressing is requested on the invitation.