- use
\usepackage{biblatex} in the preamble
- Make
biber default bibtex compiler in your editor
- load bibliography information in a separate file, say
ref.bib
- add your bibliography resource in the preamble with
\addbibresource{ref.bib} or \bibliography{ref}. As pointed out in a comment, apparently MikTeX and TeXLive handles filenames differently. So with \bibliography it's probably safer to use {ref} than {ref.bib}
- put
\printbibliography where you want it printed
- Make sure your compilation is as follows:
default compiler -> biber -> default compiler -> default compiler -> view pdf
Note that you have to run the default latex compiler twice after running biber. The default compiler can be pdflatex or xelatex, your choice. You should be able to edit your default compiler in your editor. The best editor in my opinion is TexStudio. You can add the series of commands there. Here is how you can do the whole thing automatically using TexStudio.

Here is a minimum working example.
\documentclass[a4paper, 12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{biblatex}
\addbibresource{ref.bib}
\begin{document}
Here is the reference to \textcite{holder_1889}. And here is the reference to \textcite{bouniakowsky_1859}.
\printbibliography
\end{document}
The file ref.bib contains the following:
@article{bouniakowsky_1859, series={7}, title={Sur quelques in\'{e}galit\'{e}s concernant les int\'{e}grales ordinaires et les int\'{e}grales aux diff\'{e}rences finies}, volume={1}, number={9}, journal={M\'{e}moires de l’Acad. de St.-P\'{e}tersbourg}, author={Bouniakowsky, V.}, year={1859}, pages={1–18}, collection={7}}
@Article{holder_1889,
Author = {O. {H"{o}lder}},
Title = {{Ueber einen Mittelwertsatz}},
FJournal = {{Nachrichten von der K"{o}niglichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universit"{a}t zu G"{o}ttingen}},
Journal = {{G"{o}tt. Nachr.}},
Volume = {1889},
Pages = {38--47},
Year = {1889},
Publisher = {Dieterich, G"{o}ttingen},
Language = {German},
Zbl = {21.0260.07}
}
Note: if you do not use \printbibliography no citation will be printed but the citations will be used in the document. Also, if you want to print the citations which were not referenced in the document, you have to use \nocite{*}. Here, you can replace * by a specific bibliography entry. Using * will print all the non-referenced bibliography. Here is the output.
