Someone who is mature acts grown up, like an adult. However, when one sister says to another, "That's so mature," she's usually using the word sarcastically, because her sister has done something very immature, like sticking out her tongue.
The adjective mature doesn't only describe advancing age, although the residents of a senior citizens' home could definitely be considered mature. Anything that is fully developed, like an aged cheese or bottle of wine, can also be called mature. The word can also refer to something that is only appropriate for people who are considered legal adults, like the movie that is for "mature audiences only."