There is really not an agreed-upon definition of this term, although it is used often by engineers, researchers, and politicians. In general, it refers to putting “some sort” of electronic box out on a structure that collects “some type” of data which can then be evaluated “somehow”.
One example would be a datalogger that is connected to strain sensors which are, in turn, mounted on critical structural members. Then, if an unusually high strain/stress level is detected (like from an overloaded truck crossing a bridge or perhaps a pier is tilting due to scour), the system can alert the owner. There are hundreds of other examples.
Prior to installing any monitoring system, the end user should clarify exactly what they would like to accomplish with the data, what decisions are to be made given a particular response, and what is the shortest amount of time required to answer the questions at hand. The difference in cost between a temporarily installed system and a permanently-installed system can be an order of magnitude.