Q: How do you define a 'native plant'?
A: The concept is quite straightforward: Native species (plants and animals) were those present before the advent of European civilization. Nonnative (i.e., adventive) species are those brought here as a result of human activities. It can very occasionally be tricky to determine this for a particular species (if, say, introductions took place a long time ago) but the concept is simple and straightforward.
Q: Why do you draw the line with the advent of Europeans?
A: There are excellent reasons for doing so. Ecosystems are collections of species which have coevolved over hundreds of thousands or millions of years. We may not be cognisant of all the details, however, as a result of these long associations, complex and sophisticated relationships evolved which over time have led, not to static ecosystems, but to stable and resilient ones. Adventive species are a whole other kettle of fish. Sometimes they manage to fit in (or become extinct) without apparently overly affecting the new habitats they colonize; at other times they become invasive and have all manner of disruptive consequences on native environments (for instance zebra mussels, Japanese beetles, brown spruce longhorn beetles, Asian longhorn beetles, emerald ash borers, etc.). Consequently distinguishing native from nonnative species is an extremely useful thing to do and in 99.9% of cases it is clearly and unambiguously possible to do so.
Now it is, of course, true that in times past there was some natural spread of species from the Old World to the New (and vice versa); however, such spread was (a) miniscule compared with the scale of human-mediated introductions, and (b) happened very slowly and over tens or hundreds of thousands of years, allowing ecosystems to adapt themselves to the new circumstances of one or two new species. These process are very distinct, and it is the scale and rapidity of introductions in recent years and decades that threatens to upset the ecological apple cart.
Q: How do we know which plants are native?
A: There is a whole area of science devoted to answering such questions. There are many lines of evidence that allow us to establish what our ancestral fauna is/was. There is a plethora of recent fossil evidence including the analysis of pollen found in bogs and accumulated on lake bottoms that is very helpful. Distribution patterns can also be very useful in separating native from nonnative species (for instance, nonnative species tend to be clustered in areas of human habitations and commerce, for instance around seaports, etc., whereas distribution patterns of native species show no such biases). A preponderance of adventive species tend to be found in human-modified or mediated environments (such as fields, waste ground, agricultural areas, etc.) whereas undisturbed habitats (such as bogs, salt-spray barrens, old-growth forests) tend to have very few introduced species. Genetic and DNA evidence can also allow us to readily separate native from nonnative species.
For those who are interested, excellent reading on this matter is provided by Carl Lindroth's superb book: Lindroth, C. H. 1957. The faunal connections between Europe and North America. Almqvist & Wiksell, Stockholm, Sweden. 344 pp. It is, of course, now long out of print but copies are available in libraries and can be had via resellers (such as alibris.com and amazon.com).
I wish to express my gratitude to Christopher Majka for allowing me to use his words. I also want to express my gratitude to the person who originally posted these questions to the NSNature listserve. I've shortened and simplified the original questions in order to save space, but there would have been no answers without her questions.