Mendeley originated as an open source citation manager and now partners with the publisher Elsevier to offer a free version as well as paid premium versions (including the option to pay for extra online storage beyond 2 GB). It is compatible with Mac, Windows, and Linux, and comes with plugins for Word and LibreOffice. Mendeley offers Android and iOS apps.
To use Mendeley, you'll need to create a free account.
After filling in some brief information to create a public profile, you'll end up at a page where you can download Mendeley Desktop.
Mendeley will guide you through setting up your library. Once you finish, go to the Tools menu in Mendeley Desktop to install the Web importer and Word and/or LibreOffice plugins.
Set up your online Mendeley profile:
Sign into your account from the main Mendeley page to view your library online and update your profile with your research interests, education and work history, and publications. You can change the privacy settings to control who sees your profile.
Install the iOS or Android apps to read and annotate PDFs on your phone or tablet:
The iOS app is available through the iTunes store, and the Android app is available through the Google Play store.
Click on the Mendeley bookmarklet in your browser to save citations for articles you find on the Web or in library databases, along with attached PDFs if available.
Click on the Documents icon in your library to import files/folders, set computer folders to "watch" for new files, or manually add records for sources of any type. You can also drag and drop files in directly.
Add PDFs and other file types to records by editing Details to the right of your library.
Discover new research by searching a database of almost 100 million documents, and get suggestions for sources related to your interests.
Sync and back up your data online to access it anywhere. 
Create folders. 
Star favorites and mark documents as read. 
Edit record information, including adding searchable keyword tags and attaching general notes.
Read PDFs directly in Mendeley Desktop and add printable in-text annotations, highlighting, and sticky notes.
Search the full text of your library using keywords.
Create one free private group (premium version offers more) to share citations and files with collaborators.
Showcase your own work through a public profile, get download and citation statistics on your publications, follow colleagues to see their publications, and join public groups to share bibliographies.
Insert in-text citations and bibliographies directly into Word or LibreOffice. Use the dropdown menu and click on "More Styles" to choose from thousands of general and publication-specific citation styles and dozens of languages.