Download a free demo of the game (version 0.27) here:
Unpack the file (Mac/Windows: double-click; Linux: use tar -xzvf), then double-click the game to play.
The demo works exactly like the full game, except that it doesn’t save any data to disk, so when you quit, all your designs and colonies go poof. (If this troubles you, you can find a fix here.)
If you have any trouble downloading, unpacking, or running the game, please contact support.