The library buys or subscribes to a large number of resources you can use freely; search the catalogue to discover them. At times, however, the subscriptions don't cover the item you need and you hit a paywall. You can make a document delivery request, or look for free alternatives. Open access papers in the DOAJ database or e-prints in arXiv are obvious choices, but also try the following options:
Recommended starting point. Google Scholar is a search engine for academic papers. It indexes subscription-based and open access databases, institutional repositories and personal pages.
Reading the article you need may require a subscription, but chances are that you find the free full text when you see a link labeled [PDF] or [HTML] on the right of the search result. Click "All versions" for additional sources.
BASE is a powerful search engine specialized in academic open access web resources. It is operated by Bielefeld University Library.
Asking the author has always been a popular last resort. Nowadays social networks, customized for the scientific community, connect researchers and make exchanging knowledge/materials easier and faster.
Good examples are given by ResearchGate and Academia.edu. It is worth investing a few minutes to set up a personal profile in either network, to introduce yourself to the community. Then, requesting a paper from its author becomes a one-click operation, as there is a special "Request" button for it. Sometimes, depending on the author's decision and/or the publisher's copyright policy, direct download is also possible.
Mendeley is mainly a reference manager, which has evolved into being a community of researchers. Its engine can search for papers uploaded by its members. Articles that were posted in full text by their authors are identified by a "Save PDF to library" button.