Many blend-files reference external images or other linked blend-files. A path tells Blender where to look for these files. If the external files are moved, the blend-file that references them will not look right.
When you specify one of these external files, the default option is to make the path relative. Blender stores a partial path evaluated relative to the directory location of the referencing blend-file. This choice helps when you need to reorganize folders or move your files.
With a relative path, you can move the blend-file to a new location provided the externally linked files are moved along with it. For example, you could send someone a folder that contains a blend-file and a sub-folder of external images that it references.
When relative paths are supported, the File Browser provides a Relative Path checkbox,
when entering the path into a text field, use a double slash prefix (
//) to make it so.
Relative paths are the default but this can be changed in the File tab of the User Preferences Editor.
You cannot enter relative paths into a new untitled blend-file. Save it before linking to external files.
If it is necessary to relocate a blend-file relative to its linked resources, use Blender's File Save As function which has an option to Remap Relative file links.