This page describes how to use python scripts in Blender 2.6. For information on how to use python scripts in Blender 2.4 visit this page.
To get more information on an available Add-On press the triangular button at the left of the entry. More information will be displayed like for example a description and version number. Perhaps the most useful information is location, which tells you where you can find the results of the script (where you can use it). For most scripts you'll also find a link to a wiki-page with more information and a possibility to report bugs.
If you're very happy with a certain Add-On and would want it to be enabled every time you start Blender, you have to save it to the default .blend file. Simply press the button (number 3 in the image above) and you're done. Note that this saves the current state of Blender to the default .blend file, including any meshes etc. you might have in your scene.
Further information on Add-Ons can be found at the Blender wiki.