What is nanoporous structure?
I believe this question is both simple and difficult -- especially for those who are not native English speakers. I'm going to suggest a straightforward answer, but first let me explain the difficulty.

Unfortunately, terminology related to the nanoscale is often misused. But if we can agree on what the nanoscale is, the answer to the question becomes much simpler. So I'd like to respond to this question in two parts: 1) What is nanoporous? and 2) What is a nanoporous structure? 

1) What is nanoporous?
This question relates to a more general question, "What is the nanoscale?" In my experience authoring articles in nanotechnology and serving as a referee for ACS Nano, I've struggled with this question.  It seems no accepted authority has defined the term. The closest I've found is this website which says clearly, "...the nanoscale, which is about 1 to 100 nanometers". I realize that good arguments can be made for extending this range to larger values… but surely it must stop short of 1000 nm = 1 micron. If someone knows of a more authoratitive body making a definition, I’d love to hear of it.

Unfortunately, I’ve found a significant number of scientific experts publishing statements suggesting that the nanoscale extends up to the 1000 nm range. This does not make sense to me: that would more properly be called the "microscale".  As a referee I've been surprised at the number of articles submitted for publication which claim nanoscale work when actually the objects of study are at the microscale. The motivation for this is obvious: it’s a lot more impressive to say you’re working at the “nano” than the "micro" scale these days.

So, my suggestion is that if we accept nanoscale means 1-100 nm, then nanoporous means the presence of pores in the 1-100 nm range.

2) What is a nanoporous structure? 
Strictly speaking then, a nanoporous structure would be any structure having pores in the 1-100 nm range. The term itself implies nothing about the uniformity of the pores, and that is why we refer to sponges as porous despite very different sized pores. It is true that uniformity of pore size is of great interest to scientists for important reasons, but the term itself does not include that meaning. It is a case of denotation verses connotation. Though scientists have a reputation for precise terminology, they can be just as guilty as others in applying terms imprecisely – and for the same reasons: convenience and self-interest.