The two most commonly used systems for citing references are the Vancouver system, which numbers the references, and the Harvard system, which refers to them by the name/s of author/s and the year of publication. In the first, a numbered list of references is given at the end of the paper; in the second, the list is arranged alphabetically by the name/s. This post explains a number of minor variations in citing numbered references. Incidentally, the system of numbering references is known as the Vancouver system because it was a meeting held in Vancouver in 1978 that led to ICMJE, the International Committee of Medical Journal Editors, and, eventually, to some agreement among journal publishers on a common format for references.
The Vancouver system serially numbers each source cited in a paper in the order in which the sources are cited. If a source cited earlier is cited again, the same number is used. At the end of the paper, under the heading "References," full details of each source are given next to its number.
Superscripts or in-line Citing sources by numbers seems simple enough but different journals treat the numbers differently: some print them as superscripts whereas some print them "in-line" (that is, neither as superscripts nor as subscripts).
Enclosed or open Whether printed as superscripts or in-line, some journals enclose the numbers within brackets—either square brackets, as in , or round brackets (also known as parentheses, as illustrated here).
Before of after the full stop (period) When a numbered citation appears at the end of a sentence, the number/s, in whichever form (superscripts on in-line, enclosed or open), may be placed after the full stop or before the full stop.
As the author of a research paper, you may be annoyed at such trivial differences. However, think of this whole process of getting published as a game: if paying attention to such details helps you in getting published faster, the effort is certainly worthwhile.
["Publish and prosper" is a series of posts about tips for researchers whose first language is not English but who submit papers to journals published in English. The series touches upon not only writing (spelling, grammar, punctuation, usage, and style) but everything else relevant to publishing research papers that journal editors wish their authors knew.]