Although this information can be found elsewhere on the web, I still feel its useful for me to highlight software that can be used for free that replaces popular (pay for & Non pay for) software.
Web Browser: The commonly used one (and it is free to use) is Internet Explorer, in my opinion however it isn't the best, try Mozilla Firefox, Opera or Flock. They all have their own advantages, see also the article on flock (my personal favourite) on this blog.
Email Application/PIM: The common one is Outlook Express (Free) or Outlook (Part of Microsoft Office, i.e. Not Free!). If you are using Outlook Express at the moment then Mozilla Thunderbird is a drop in replacement and better. Outlook has personal information manager (PIM) features aswell, these can easily be accessed via Mozilla Sunbird, which is a standalone PIM application. For linux users Evolution is another option which has both Email and PIM features in one application.
Office Suite: The standard examples are Microsoft Office, Microsoft Works etc. An excellent alternative is Open Office, which features all of the common applications of Microsoft Office and can import Micrsoft Office files. For Linux (especially KDE users) there is also K-Office.
Instant Messaging: Standard examples are MSN messenger, AIM, Yahoo Messenger, in windows it is best to use these native applications, in Linux Gaim (for Gnome) & Kopete (for KDE) will allow you to use ALL of your chat accounts in one package!
Website Creator: Perhaps a more advanced tool than those listed above but it seems many people do this, the standard example is Dreamweaver, a free alternative that will suffice for the vast majority of users is NVU which works in both Windows & Linux.
Image Editor: For playing about with photographs, the standard example would be Adobe Photoshop and slightly less feature filled (and Cheaper!) Corel Photo Paint. A completely free example would be The Gimp which really is excellent. Also for Photoshop users try GimpShop, which makes The Gimp look more like Photoshop.
Drawing Package: Standard examples would be Adobe Illustrator & Corel Draw. A very good easy to use package for all but the most advanced users of the above would be Inkscape.
Audio Editor: Standard example would be SoundForge or Cooledit. An excellent free alternative is Audacity.
Note: This is not supposed to be an exhaustive list, just a basic start, there are much longer lists around, one of which is linked to below.