Google Drive (formerly called Google Docs): the collaboration features are fantastic.
Take a look at this video and this one on sharing/collaboration.
Here's a good article on How to Make Google Drive Work Like a Desktop Suite.
Mobile-based:Quickoffice (Android, iOS)
Kingsoft (Android, iOS)
Documents To Go (Android, Blackberry, iOS)
Office^2 HD (iOS)
Apple iWork (iOS)
Desktop-based:Many of the packages listed here work on Windows, MacOS and Linux. Not all of them work equally well on "supported" platforms, so be sure to test the out on all of the platforms of your choice.
LibreOffice is the most popular alternative to Microsoft Office. It is free, you don't have to pay anything to use this. It works on Windows, MacOS and Linux.
LibreOffice 4.0: First Take
LibreOffice 4: A new, better open-source office suite
LibreOffice 4.0 New Features and Fixes
You may also wish to check out:
- IBM Lotus Symphony
- Ability Office
- SoftMaker FreeOffice
If you have needs that are more like desktop publishing you might want to check out Scribus.
Microsoft has a fairly limited notion of what needs to be in an office suite. Calligra contains a number of interesting additions, also, consider some of these:
- Alchemy / Webchemy
- Open Clip Art Library
Here are some other collections of this sort of thing:
40 Best Open Source Graphic Programs
10 Excellent Open Source and Free Alternatives to Photoshop
YouTube is a great place to learn how to use these things well, here are a couple of examples:
Design Professional Brochures Using GIMP Inkscape and Scribus
GIMP Tutorial - Fire Effect
Helpful Article for what to do/not do when switching from Office:
Triumph and disaster: Two migrations to OpenOffice
Good comparison article, be sure and check out the comments.
Battle of the Office Suites: Microsoft Office and LibreOffice Compared