Mendeley is a relatively new tool that can be a little difficult to describe succinctly.
Similar to traditional citation management programs, like Endnote or Reference Manager, it helps collect citation information; keeps your research in a central, organized place; and makes it easier to insert references into your academic writing. In addition to these so-called "citation management" tasks, Mendeley also has drag-and-drop functionality for importing PDF files into your Mendeley database. That means that if you've already collected many (or a few) articles, and you're looking for a way to organize your efforts, Mendeley can help.
The same is true if you've just found and downloaded a new PDF file and want to be sure not to lose it in your own maze of a computer. Mendeley also has workflow tools such as annotations, tagging and full-text PDF searching, which allow searching and categorizing to facilitate your thinking and writing around these files. It also allows users to share citation information and PDF libraries between collaborators. (Something like: "You uploaded and are reviewing this paper...other people who also uploaded it are reading these other papers...")
It's available free of charge to anyone with internet access. (This is a so-called "freemium" service that offers basic--and arguably very adequate--access free of charge but offers other premium features for a monthly charge).
Troubleshooting & help
Reference librarians can offer troubleshooting over the phone (212-305-3692), via email (email@example.com) or in person at the Reference Desk.