In order to save the game in Horizon Zero Dawn, you must first locate a campfire. It is not possible to save the game whenever you wish, so you must locate one of these if you wish to save the game manually. There will be periods of the game when you trigger an event in a quest that will cause the game to auto save. The auto save will also store your progress, allowing you to continue at a later date. However, for manual saves, campfires are the way to go.
The screenshot above shows a campfire. If you locate one, it may not be lit. You will first need to light the fire so you can save the game. Once the campfire is lit, you will be able to save your progress. Walk up to any unlit campfire and press triangle to light it.