Often people just creates lists for replacing software from Windows to Linux or OS X, in this list I will write about Windows alternatives too! Enjoy!
The most used is always Microsoft Office, why? Because it works actually very good! (and most of people use it, so we need to open it though the official suite to look it correctly). Although the licence costs are high and the pseudo-standards change way too much between versions, making impossible to look it correctly while not using the same version where it was made (or newer).
There are lot of alternatives, freeware, open source suites and other proprietary alternatives and the correct one for you depends on what you need, if you want a word processor, all of them will make the job correctly, if you want presentations probably you will like iWork and for spreadsheets Microsoft office is the only way.
Personally, I like Microsoft Office and KOffice (now Calligra both done by KDE.org guys). The freeware alternative is IBM Lotus Symphony, for open source is LibreOffice (based on the other open source OpenOffice.org) plus, this last is supported by the OpenDocument Foundation. If you want propietary software, Microsoft Office and iWorks (macfags will love) are the way, both are pretty nice BUT only works on OS X and Microsoft Windows. On Linux the way is LibreOffice if you need compatibility with Microsoft Office and you want a powerful office suite, and if you like Qt and KDE — i’m sure you want all your applications running under it, so KOffice (Calligra) is the way. If you want it to be multi-platform, plus cloud computing, Google Docs will do the job!
This part is the most conflicting to me. I love music, I need an equalizer but also I need that it works pretty fast with a very big music collection. The operating system that got lots of GREAT software for this is Linux. The selection of this will MOSTLY depends on that if you got an iPod or an iPhone. If you got one of this gadgets on Windows or OS X, the way is iTunes, if not, probably Foobar2000 (Windows) or Songbird (although both got iPod plugins, is not the same and we know that). If you’re running Linux, there are lot of options, but the better ones are Amarok, Banshee, Clementine (based on Amarok 1.6, works on OS X and Windows too) and Rhythmbox, all got iPod support.
Most of people uses Nero or Roxio software because always comes with the computer, the problem with this suites is that is pricey plus, bloated and slow. There are lot of superior alternatives, alike ImgBurn for Windows (which is very cheap), for OS X you don’t really need one if you know how to use Disk Utility, but there is SimplyBurns with is good and a BIG suite (alike Nero) that will do any kind of works with CDs, Toast. For Linux k3b is the only and best way!
Since lots of years ago (I think from Windows XP) Windows haves this feature already build-in, the same way on Mac OS X and unzip at Linux, but theres a lot of people that uses WinRAR to compress their stuff, for Windows theres a opensource alternative, 7z which (un)compresses RAR, Zip, plus 7z at Windows. For Linux and OS X you can install WinRAR command to the job, and for other kind of compressions there is tar!
This mostly is a personal election, just DO NOT USE INTERNET EXPLORER IF YOU ARE NOT USING IT TO DOWNLOAD ANOTHER BROWSER. Personally I like Opera and Chromium (Google Chrome). But there are other good alternatives like reKonq and Firefox. The final option will depend on the extensions you want.
Messaging and Videocalling
Sadly we need to use the protocol that other people use, and the most used ones nowadays are Windows Live Messenger and (worst thing ever) Facebook. But we can make our experience way more comfortable if you got the right applications. For Linux there are a lot of alternatives that got a plus, connecting with multiple accounts on different services at the very same time as Pidgin, Kopete and Emphaty and for the people that likes WLM-a-like GUI, there is KMess and Emesene. For Mac the best thing, and the one I use is Adium.
For videocalling there are not a lot of alternatives, the best one for me is Google Video Call which is way better than WLM. But if you want hi-res an application that lots of people uses, there is Skype. For conferences the best is Google+ Hangouts!
Mobile messaging and calling
I’m tired of Blackberry Messenger users! They thing that everyone needs an outdated phone with an outdated operating system (this is a personal opinion) but there are ways to communicate with other smartphones via messaging alike BBM. The one I use its Whatsapp which is multi-platform, but there is Google+ Huddle (Android and iOS), Pingchat (multi-platform).
Plus, for calling. There is Google Voice, Google Talk, Tango, Skype, Viber, this last one is the one I like. But just Tango, Google Talk and Skype got videocalling.
I think its all, if you want more just poke me at the commentaries and I’m writing other alternatives for different uses.