Citation management software are programs that help researchers organize their citations and insert them into their papers. Often these tools provide for "cite while you write" capabilities with add-on software compatible with Microsoft Word or other word processing programs.
Typically, you download citation management software to your computer and run it. You may need administrative permissions for installation on shared computers.
Zotero is a great choice for a free and user-friendly citation management program. It is compatible with Mac, Windows, and Linux.
One of Zotero's coolest features is its Chrome Connector, which allows you to save citations from webpages with one click.
Mendeley is a great citation manager that offers a lot of similar features as Zotero. It has a desktop app, cite as you write capabilities, a web connector for one-click saving, and is also available for Linux, Mac, and Windows systems. In addition, you can create public or private groups to collect citations on a topic with other researchers.