In a math paper, what is a remark?

I sometimes see paragraphs labeled 'Remark.' However, papers that include remarks also include unlabeled explanatory paragraphs (i.e. all the other writing in the article) that seem to be remarks. What exactly is a remark?

The remark device is used for material that is is too long to be included in parentheses, while it deserves greater prominence than a footnote. Typically, there is a point in the text where a certain issue is prompted to arise in the author's mind—or the author expects such issue to arise in the reader's mind—and the author feels the need to get it out of the way before it becomes a distraction.

For example, the author may introduce a particular definition or notation following a certain convention, while some other authors follow a different convention. From a strictly logical perspective, the author does not need to remark this: the reader simply has to follow the given definition. However, the reader might have encountered the differing convention elsewhere, and could be confused if she does not make a specific note of the difference in the present work. It is therefore a courtesy to the reader to point out the differing conventions, and perhaps explain why the author chose the one used in the present text. This explanation is put aside in a remark, which can be skipped through by the reader who does not need it.

Another use of a remark might be to give the reader a slight break by introducing some interesting and colourful information (e.g. a historical anecdote) at a point where it is appropriate—albeit not logically relevant to the developing argument at hand.

Remarks may also be used to provide motivation, for example by citing applications, or to point out certain pitfalls in understanding that might trap the unwary reader, or perhaps to explain why the author is taking what may seem like an unnecessarily circuitous approach at a certain stage.