Word/OpenOffice/LibreOffice/NeoOffice
If you want to use Zotero with one of these programs, you have to start by installing the required plug-in. When you have done this successfully, a Zotero menu will show up when you open your text editor.
Once you have found your reference, added it to your library and are ready to cite it, you choose the Zotero menu. Make sure that your pointer is located where you want your reference and choose Add/edit citation. This will prompt a red search field where you can access your library and search for you reference. When you have chosen your reference and pressed enter, the reference shows up in your document.
If you want to change the citation style, choose Document preferences from the Zotero menu. Once you have inserted one or more references, you also need a bibliography. This is inserted by choosing Insert bibliography from the Zotero menu.
You can find videos explaining how to use Zotero with both Word and Open Office here, and explanations with pictures are availble in the guide Mastering Zotero.
LaTeX with BibLaTeX
You can use Zotero even if you write in LaTeX and use BibLaTeX for references because Zotero easily lets you export your references. To do this, you choose File -> Export library and choose BibTeX as your format. The file you get has .bib extension and can be used as your bibliography file with LaTeX/BibLaTeX. Read more about how to use LaTeX/BibLaTeX in this guide by Knut Hegna and Dag Langmyhr.