I mean that question quite um…seriously.
Paul Krugman, who I believe originated the concept, recently defined it as follows: “…someone distinguished by his faith in received orthodoxy no matter the evidence.” I would rather have something less normative and also more specific.
I think of it this way: the People are Very Serious if they realize that common sense morality must, to a considerable extent, rule politics. At least if voters are watching.
So what is common sense morality in this context? It embodies a number of propositions, including, for instance (with cultural variants across nations):
1. Political decisions should be based on what people and institutions deserve, based on their prior conduct and also on their contributions to the general good.
2. Economic nationalism.
3. Traditional morality, based on respect for authority, repayment of debts, savings, and hard work.
4. Inflation is bad, in part because it violates #1 and #3, and in the case of the eurozone it often violates #2 as well.
5. “I don’t care what you all say, the government should be able to find some way of arranging things so that I don’t have to suffer too badly from this.”
Now here’s the thing: common sense morality very often is wrong, or when it is right that is often with qualifications.
Therefore at the margin there is almost always a way to improve on what the Very Serious People are pushing for. The Very Serious People realize this themselves, though not usually to the full extent, because they have been cognitively captured by their situations. They see themselves as “a wee bit off due to political constraints,” instead of “a fair amount off due to political constraints.” So there is usually some quite justified criticism of the Very Serious People. Common sense morality is needed at some level, but still at the margin we wish to deviate from it.
That said, it is a big mistake to try to throw the Very Serious People under the bus. The Very Serious People understand pretty well how to deal with a public which believes in some version of 1-5, and furthermore they often know that such public beliefs, whatever their limitations, are useful too. Anyone else trying to manage the situation may come up with some favorable breakthroughs, but also may make a total hash of it, as was the case with Syriza. Syriza failed to realize the import of 1-5 for both domestic and foreign politics,and so they drove the Greek economy to the point of total desperation. There is a nested game going on, where the public has a big say on the heavily publicized issues, and the politicians must in some way heed that.
If you want to try the “replace the Very Serious People” game, and assume the subsequent risks, that is a judgment which can be made. The mistake is to think that the partial wrongness of the Very Serious People is necessarily a reason to take matters out of their hands.
Addendum: Via Tim, here is the entry on Very Serious People on RationalWiki, it seems Atrios first put forward the concept. And here are the remarks of Tsipras.