If you have the computer play for you, the computer will develop its own strategy, just like regular computer opponents. This is a good way for novices to learn. It's also very convenient when you're either about to win (or lose) but don't want to finish the game yourself. Select this option, too, when you have to leave a multi-player game early but don't want your territory to fall completely apart. (You'll have to leave the game running.)
Having the computer play for you still leaves you some control. The computer will never change your tech spending bars. If you build ships, the computer will use them. If you send ships somewhere, the computer won't redirect them until they reach their destination. The computer will generally try to keep the colonies you have, unless it believes it's spread too thin.
If you select "just mark my turn done," that's all that will happen. None of your budget bars will ever change (except when you lose or gain planets). This is a convenient option when you have several turns in a row where nothing interesting is happening, like when you're waiting for a colony to become profitable.
The scroll bar at the bottom of the Auto Play dialog determines how long each message is displayed. Normally, you have to click each message to make it go away. Under Auto Play, you can still click, but if the time runs out, it will go away automatically. This is only important when you're watching the computer on Auto Play.
You can set the style of the computer play when you put it into Auto Play. The scroll bars in the Auto Play dialog set how much your computer will prefer to attack and defend. Play with the controls -- you'll figure them out.
You can have auto-play turn itself off when something "interesting" happens (like a battle or something). When a turn updates if you hear "Check this out!" auto play has been automatically turned off and you should see why.