When talking about a citation method, we mean the way of writing citations in the text and the form of bibliographic references in the bibliography. Citation methods do not determine the basic way of referring to a source (i.e. whether to use a digit or the surname of the author and the year for publication) but they specify the ordering of bibliographic references (chronologically or alphabetically).
The instructions of the respective citation style tell you what citation method to use when citing. Some of the citation styles allow you to choose from multiple citation methods. Therefore, it is always essential to read the instructions of the publisher or institution carefully so that you know what citation method is required. Authors publishing in scholarly journals may even find themselves in a situation where the instructions for citing do not specify the citation method. In this case you need to determine what citation method to use from articles published previously in the respective journal.
Remember: a citation method is not the same as a citation style
Citation-sequence method is often used in biomedical and natural-scientific fields. In this method, citations take the form of order numbers referring to the bibliographic references, that are listed in the bibliography as they appear in the text. In the event that a citation repeats, the same number as before is used.
The individual citation styles may use various forms of citations. In the example here, the digits are written in superscript, but other styles may require writing the digits in round or square brackets, etc.
In the biomedical field and natural sciences, you may encounter the citation-name method. Here citations are again marked with an indexed number that refers to a bibliographic reference listed alphabetically according to the surname of the first author. When citing the same source again, the digit assigned earlier is used.
Furthermore, here too the individual citation styles may require various forms of citations in which the digits may be in square or round brackets, without brackets altogether, etc.
If you do not use one of the citation managers, it is necessary either to have a final list of literature prepared before you start writing the text or enter the digits into the text after you have finished writing it.
The author-date citation method (sometimes called the name-year method) uses the form of citation surname of the author and the year of publication. The bibliography is then ordered alphabetically according to the surname of the first author.
If you cite multiple works published in the same year by the same author, usually the letters a, b, c are added in both the citation and bibliographic reference to the year of publishing unless specified differently by the citation style.
The individual citation styles then specify whether only the first author or others as well should be mentioned in the citation if the work was written by multiple authors; whether the bibliographic reference should contain the year of publication in brackets or not; whether in repeated bibliographic references the details of the authors should not be replaced by e.g. a dash; whether to list bibliographic references chronologically, etc.
Especially humanities and social sciences use the footnote method which consist in writing numeric citations referring to bibliographic references placed on the same page in a footnote. If you cite the same publication repeatedly, a new order number is used. Individual citation styles may then require different forms of bibliographic references (e.g. ISO 690 allows making a reference to an earlier assigned footnote number with the whole bibliographic reference of the respective document instead of repeating it).
The author-title citation method is seen only very rarely. In this instance you use the surname of the first author and a maximum of three words from the title as a citation. The bibliography is then ordered alphabetically according to the surname of the first author. Due to the rare use of this citation method, a real-life example could not be found and therefore the example below has been created purely for illustration purposes.