Apple uses QuickTime to play videos. This software is built in to the Mac OS X operating system. The "Software Update" under the Apple menu will make sure you have the latest version. Adobe Flash no longer comes preinstalled on new Macs, but you can get it from Adobe. Windows Media is the other major format, and that definitely requires additional (free) software. Very few people use "Real Player" anymore so I would not install that.
If you need Flash, go here:
http://www.adobe.com/support/flashplayer/downloads.html
Flip4mac lets you play Windows Media files on your Mac. You can get the installer here:
http://www.apple.com/downloads/macosx/video/flip4macwindowsmediacomponentsforquicktime.html
I also recommend Perian, which adds additional media types for QuickTime.
http://www.apple.com/downloads/macosx/video/perian.html
One more thing: VideoLan (VLC) can play almost any form of video. It's free. Download here:
http://www.videolan.org/vlc/download-macosx.html
Movist also gets good reviews for playing just about anything. Another freebie: