Mature men are men over 40 or sometimes over 35.
Older men, generally over either 35 or 40 depending on your own definition, are considered sexy by many men and women. The people attracted to mature men generally like the fact that they are sexually experience, know what they want and how to get it. Many also like the physical attributes of older men including salt-and-pepper or silver hair, weathered face, etc.
Historically husbands were years old than their wives, often the same age as their wives' fathers. This was at least partly because men got married after they were well established financially and were able to support a wife whereas a wife with the most childbearing years was considered desirable.
Mature men are well appreciated in parts of the gay community, and the bear community and leather community have a large older presence. Older men, often called "daddies" in gay slang, are perceived as more stable than younger men, which is appealing to some men and women. Older men are also considered more patient during foreplay as younger men are considered to eager to get to the intercourse. Also people into age play who wish during role play to act younger than they are may find that doing this with an older man is more satisfying.