Today, most academic libraries subscribe to hundreds of research databases, and will generally have a list of databases organized by subjects.
For example, here's the Vanderbilt Library's Find articles & databases page.
A library's catalog lists items it owns: books, journals, music scores, paintings, laptop computers, etc.
A database contains records, sometimes full-text articles, sometimes citations to articles. It is often subject specific and is usually produced by a third-party firm.
Your search in an article database will usually return either a full-text article or a citation.
If your search results include links to full-text articles, you can print, email or download your article to a USB/flash drive.
Be aware that if you email your article, it may only send the citation and a link to the full text. When you click on the link from off-campus, you will probably get a log in screen that asks for your university account information.
If your search returns citations only, you may still find the article somewhere in the library. At some institutions, you may see a special button in your results you can click to link to the full-text of the article online.
The next step would be to check the library catalog to find out if there is a print subscription to the journal. You will need to photocopy the article in this case. Most libraries provide self-service photocopying where a library user pays to make the copies him/herself.
Try Interlibrary Loan (ILL)!
Academic libraries can usually borrow items such as books and reprints, scores, proceedings, etc. from the circulating collections of other libraries.
Microforms, newspaper back files, dissertations, and government documents often are available, but may need special handling. Periodical volumes, reference books, rare or fragile items, videotapes, and very old or very new imprints cannot normally be borrowed.
Your library can also request copies of journal articles and many other items that cannot be borrowed. Photoduplication service is subject to copyright laws and guidelines for fair use of copyrighted materials.