There are a few quality answers already (I specifically second Peter Jansson's answer), but I wanted to give a more practical perspective.
Are we expected to cite every reference relevant to our research
This would not be practical, or realistic, in many fields. I for instance work with data analysis in cancer research. The amount of "relevant" work published every month is absurdly large, so much so that even if I read and only read, everyday I would not be able to maintain a thorough grasp of the literature.
In some other field, where the problem formulation is well-framed, and the boundaries more clear-cut, this might be a more feasible expectation.
Are we expected to cite only the “best” references relevant to our
research article, where best can be defined as the papers which have
appeared in the best journals or papers that are most cited?
Interesting you take up this question of "best articles" vs articles in "best journals". There is some selection bias here, also known as "rich get richer" phenomenon, giving rise to a Power Law distribution among citations.
Articles published in high impact journals usually reach more people, tend to be written by more renowned scientists in the field. Also people have a tendency to find new reading material based on the number of citation an article gets. Alternatively, the chance of "finding" an article is increased by the number of other articles citing that paper. So the more renowned a particular study, the more renowned it will get.
There's little you (or I) can do about that, sadly.
Do neither of the above apply? Are we simply expected to cite any
paper relevant to our research article, just as long as we have cited
something that supports/is related to our argument?
You should cite the papers where you get your prior information from. Simply put; you use a particular finding from another article, you cite them. Using prior knowledge might be (and usually is) applicable in multiple scenarios (note that below is not an exhaustive list):
- You help make a case in your introduction: e.g. by giving statistics, or portraying the current state of the field (a.k.a paying your dues)
- You refer to a particular method, protocol, instrument etc in your Materials & Methods
- You back up your findings, or their implications, by citing similar findings by independent researches supporting your findings (or interpretation of them), typically in Discussion part of your paper.