Application directory
The application directory stores
The directory structure is self-explaining.
You can delete for example the configuration-file; it will be generated new with the defaults.
It is recommended to create backups from this directory, before you install a new version.
Installation by Zip-package
Zip package is just one Factorio directory that doesn't need to be installed, all the saves are there locally easy to find.
Installation via Installer
Windows
Installation package follows UAC. For that case the factorio directory is where you install it (Program files by default) and the user data (saves, player data etc) are stored in
C:\Users\<username>\AppData\Roaming\Factorio
Windows Vista and newer (UAC active):
%appdata%\factorio
contained
MacOSx
The application should be by default installed in the Apps-folder.
Whereas the data is stored under
/users/<replace with your username>/Library/Application Support/factorio
Which is equivalent to
~/Library/Application Support/factorio
You can type in this into the finder (press SHIFT-COMMAND-G for goto folder) and put in exactly this line.
If you still don't see this folder it might be invisible for you. Search in the internet for for example about "make library folder visible macos".
Example link: http://finerthings.in/featured/lion-library-folder-visible/
Linux
Data is stored under
~/.factorio
Changing the save directory
It is possible to change the save directory for Factorio. This can be done by editing the location of the "write-data" entry in the config-paths.cfg file, though this also affects the location of the other directories (mods, scenario, temp). If you want to move only the location of your save files you can change the the save directory using symlinks.