These old versions are unlikely to work for OpenStreetMap editing.
Mac: Simply download the application and click "Open".
Windows: Download the Windows installer and open the file. If you're warned about an unsafe file, click "More info" and "Run anyway". Continue through the installer, agreeing to install Adobe AIR when prompted. At the end, Potlatch will open. A shortcut will be created on your desktop.
Linux: You can use Wine to run the Windows installer.
Windows application: This won't install the app into Program Files. Download and unzip the application folder (don't try and run from the zipped folder). Navigate into the folder and launch the Potlatch file. If you're warned about an unsafe file, click "More info" and "Run anyway".
Mac or Windows AIR file: You can download AIR from Harman.com. Once you have installed AIR, you can just open the Potlatch.air file.
CPU rendering versions: The standard AIR file uses GPU rendering. If you have problems with dashed lines and text appearance, you may want to switch to the CPU rendering version instead.