Mendeley allows you import full text articles and store them in your account. Any open source/free download will work using the import PDF option in the browser plugin.
However Mendeley is not always able to directly import full text directly from paid databases which are behind an access wall, i.e. most of our databases. Though it is possible to download the full text from a database to your local computer then upload it to Mendeley.
If you're using a website which provides open access to its articles you can use the browser plugin. Click the browser plugin at the article level and make sure the "Download PDFs if available" option is selected and it should bring in the PDF and the citation in to one record.
Sample web browser plugin.
Desktop App entry with attached PDF.
First, download the article to your local computer. Then from the Mendeley Desktop App click the Add New >> File(s) from computer option in the menu . Select the file you would like to upload. Mendeley will bring in the file and create a citation at the same time.
Make sure to check the citation for accuracy. The quality of the citation is only as good as the hidden metadata attached to the file. It is possible to import the citation and the file separately then merge them later if you find the citations created by the file upload process are inaccurate.
The following video shows alternative methods of importing articles in to Mendeley.