A thirty-something is a person in the age group 30 to 39 years old. As in the case of twenty-something, no single word exists in the English language to denote this group or its members, in the manner of quadragenarian to mean a person 40 to 49 years old, quinquagenarian for 50 to 59, and so on.
Both "thirtysomething" and "twentysomething" appear to have entered general usage in the late 1980s, when persons who happened to occupy those respective age brackets in Western societies in general, and the United States in particular, were thought to have been shaped by fundamentally different experiences, resulting in contrasting basic philosophies and worldviews. The origin of "thirtysomething" in this context — and very soon thereafter, "twentysomething" in back-formation — appears to stem from the title of an American TV series — thirtysomething — that debuted in 1987 [1].
Thirty-somethings of that era were generally seen as possessing an outlook on life colored by the loss of the perceived freedom that characterized their twenties (often exaggerated) due to the pressures of providing for their children and holding down demanding jobs to finance the family lifestyle (they tended to marry later than those who came immediately before). Their material tastes were often identified as a mixture of nostalgia for their youth (Howdy Doody, etc.) and a continuing battle to avoid appearing middle aged while increasing the material comfort of their lives — a mindset often denoted by the informal term "yuppie". They were regarded, by both themselves and others, as the youngest cohort of the Baby Boomers.