If you're a newcomer, you might be wondering how to save the game. Thankfully, it's very simple.
Elder Scrolls Online: How to save
As per the game's support site:
There is no need to save the game. The Elder Scrolls Online is a Massively Multiplayer Online Role Playing Game that constantly saves your progress automatically. When you log out of the game and log back in, you will pick up where you left off.
That clears things up nicely, and also gives you one less thing to think about when turning off your platform of choice.