This adjective means the opposite of playful or drunk. When attending a funeral, you are expected to be sober in both senses.
The most common meaning of sober is "not drunk" — people who drive need to be sober. Sober sounds a lot like somber, and it often means sad and quiet too, or sometimes too serious. Another meaning is to be logical or realistic about something. Sometimes sober means drab: a grey station wagon is a more sober vehicle than a hot red convertible. All of these meanings are quiet and restrained — unlike a person who has had a lot to drink.