• works with most browsers (all, via somewhat limited Zotero bookmarklet)
• Works with wide variety of content types, including websites
• Saves PDFs and screenshots ; can attach any sort of online file (although may not be searchable)
• Saves bibliographic information, notes, tags, and related items
• Searchable - no more silos
• useful after you graduate - retain everything, add more 

Web Access:

• Optional
• Allows you to access your Zotero Library online, and sync your items between different computers
• Can just save bibliographic data and notes (free) or add attachments like screenshots and PDFs ($)
• Basic citation generation
• Can add or edit online (but better to use the Zotero Bookmarklet if you are adding multiple items, altho it doesn't work with all source types)
• Can use Zotero's group functionality (...you might be able to use this somewhat just via a Zotero account...)
• Latest update makes it look just like your desktop/laptop Zotero interface

Third-Party Tools:

Zotero's open source-ness has created a pretty rich ecosystem of applications and tools that work with it. Here are a few, and you can find more on the Zotero Plug-ins page.

• 2/19 - Do you have an Android phone or tablet? Try ZotEX2 (find tutorials under the blog link, and here (link) is a basic recap of functions from the developer).#Requires use of online Zotero
  • 4/2020 UPDATE: I just found a new Android app, Zoo for Zotero. Have not yet checked it out myself, but I will.
• Do you have an iPad? PaperShip has a great Zotero interface (2019 update: maybe not anymore...I'll look into it and post an update) #Requires use of online Zotero
• Do you have a tablet of any kind? Just using Zotero.org works well, and you can open, read, and annotate your PDFs in PDF-reading apps like iAnnotate or PDFExpert  #Requires use of Online Zotero.  (Alternatively, ZotFile creates a nice workflow for moving your PDF to your tablet app to annotate them, then moving it back.)
• ZotFile - extracts PDF annotations into Zotero Notes, works with online storage site like OneDrive, Google Drive; moves articles between a tablet and your Zotero library for easier reading and annotation; and more.
  • Demo screencast on installing it and using it to extract PDF annotations to Zotero Notes (YouTube ; not transcribed yet)
• Zutilo batch edits tags and much more, including making Relating items easier and making moving from edited volume to chapters and vice versa easier.
• ....and more!

I'll be adding more to this - I'm trying to make a series of shortish (around 20 minutes) screencasts that demonstrate Zotero.

Jason Puckett’s Zotero page (Georgia State) https://research.library.gsu.edu/zotero (Jason has also written a book, updated in a 2nd edition, Zotero: A Guide for Librarians, Researchers and Educators)

Hannah Gascho Rempel’s Zotero tutorial (Oregon State University) https://guides.library.oregonstate.edu/zotero

Getting Started and Installing Zotero (University of Michigan) https://guides.lib.umich.edu/c.php?g=282995&p=1885561

Migrating from EndNote to Zotero (Zotero.org) https://www.zotero.org/support/kb/importing_records_from_endnote

Migrating from Mendeley to Zotero (Zotero.org) https://forums.zotero.org/discussion/63737/moving-library-from-mendeley-to-zotero

Using Zotero with LaTeX/BibTeX (MIT) : Programmable Search Engine (google.com)

As a open source tool, Zotero has a robust number of third-party add-ons (plug-ins) that really make it even more useful. You can find a (not comprehensive?) list on Zotero's Documentation Page.

I have started some demo videos for these below.

Here are some additional  videos (some I created, some from Zotero's documentation, that may be of use.

12/22/17: This was created as an aid for two webinars I did for WSU's Global Campus. (Links below). It was before Zotero's big 5.x update, so this will be slightly out of date, although mostly still usable.

Zotero Part 1: Getting Things into Zotero:

https://www.youtube.com/watch?v=lTCdsB5iSRw

Zotero Part 2: Getting Things Out of Zotero (and into your research papers!)

[this one was not that good :-(  I was really tired and it shows, especially when I was showing how Zotero works with Microsoft Word. I'll probably do an update and post it here.]

https://www.youtube.com/watch?v=M083u7zhIuE

8/2018 update

• A two-hour screencast recording of a class I taught in the fall of 2017: https://drive.google.com/file/d/0B0AA7F-_qOc6eDdtdHZpWUFBbVk/view
• I would also recommend this really excellent online tutorial from Oregon State university Libraries: http://guides.library.oregonstate.edu/zotero