Make a postcard
While in the game, find somewhere nice or interesting and press F9. On Windows, this will save an image to Documents/Proteus/Postcards, while on Mac it will save to Proteus/Postcards in your home directory (~/Proteus/Postcards).
(Currently animals will be not be exactly captured by postcards, so I'm afraid you'll have to wait until the next update to share postcards of creatures doing funny things)
Send a postcard
Attach the image to an email, upload to the web, whatever you would do with any other image. The important thing is to make sure the patch of random-coloured pixels in the bottom left is intact. Scaling the image or converting to a jpeg might break this. The game doesn't care about the size or content of the image, as long as those pixels are present.
Open a postcard
You can only open postcards after playing through once, but you can shortcut this by clicking on the leftmost button in the top right. This is because we'd prefer new players to play through "normally" but we're not going to be strict about this. Once postcards are available, you can view them by clicking the down arrow that appears at the bottom of the screen. (This interface may be refined in future updates)
Any postcards created by you will automatically appear here. If you want to see postcards from other people, you just need to copy them into the Postcards folder.