What does "Publish or perish" really mean?

For graduate students, it means "Publish or you won't get an academic job." In many parts of academia, not getting an academic job is considered equivalent to death.

For assistant professors, if means "Publish or you won't get tenure." In many parts of academia, not getting tenure is considered equivalent to death.

Lastly, is it possible to be a researcher without concerning about "Publish or perish"?

Sure, of course.

If you're publishing.

First a disclaimer: I personally do not agree with some "ways of life" I am going to describe and already upvoted other answers :-).

is it possible to be a researcher without concerning about "Publish or perish"?

YES, albeit probably only in atypical settings.

• firstly, at this site, we tend to forget that academia is not only the first league of the few top-notch research universities, but includes a MASS of smaller universities and research institutes in all sorts of small, hidden corners of the world which tend either not to produce academic output in terms of journal articles, books, conference papers, etc., but at which they live and breath by e.g., primarily education, local politics, etc. Yet, on paper they claim to do research, so working there, you would be officially a researcher. I am speaking for instance about universities in countries, where rigorous science and high education, for whatever reasons, does not have a very strong tradition. At such places, doing research would resemble a kind of a cargo cult. Most often, at least at some stage in the career, you still need to publish something. E.g. a dissertation would probably suffice. But often an interview with such a researcher in a local newspaper would count at the place more than a first-class academic journal. Being coined an expert on X by the local media a single time would allow you to survive at such a university for a decade (at any level from a PhD student to a Full Professor) without being concerned with perishing. If there is a desperate lack of teaching staff, then you do not have to care even for being any good teacher either and you wouldn't perish. I know personally people who are are doing some research (or at least everybody around says so) for decades without moving from a place and without publishing even a technical report and do not perish.
• to a more optimistic note, though being pedantic now, you can easily be a researcher and not publish in the industry. Many industrial researchers do not primarily work for the benefit of the humankind (as you could see said in academia), but for the benefit of a company.
• finally, I speculate there could be some special professorship positions where you do not have to worry about publishing anymore, because you are not about to perish anymore.

I will add to JeffE's answer by stating:

For researchers, it means "publish or you will not get more funding". Funding is evaluated based on the strength of a proposal but if you are not showing a strong regular publicationr record, the interpretation might be that "you use money but do not produce", hence a risk not worth taking.

Lastly, to answer JeffE's question (thereby emphasizing his implicit statement): No!