The rulebook picture above isn't accurate enough..
Push out is a special shot that can only be played after the break in 9-ball and 10-ball.
After the break you have the option of calling a push out (EVEN IF you can see the lowest number ball, it doesn't matter) and shoot the white however you like. By this I mean that you can hit the white and make contact with another ball other than the lowest number ball (or even with the lowest number ball, it's up to you) or you can shoot the white and not make contact with any ball, even with any rail. Basically you can do whatever you want besides potting the white, or intentional miscue (in both cases it's a foul and ball in hand for the other player)
When you play a push out, the opponent gets on the table and sees if he/she wants to take the shot. By this point on the rules are "normal", that is that he has to hit the lowest number ball and make contact with a rail or pot something. If he doesn't want to take the shot, he turns the shot back to you, and then you have to play on, you don't have any other option.
As I said, the rulebook above isn't accurate. You can play push out even if you see the lowest number ball. It's all matter of your strategy.
If you are playing with a 3 point break rule (as all major tournaments in Europe as far as I know) there is an additional rule.
Let's say player A breaks and it's an illegal break (no 3 points achieved), then player B gets on the table. Player B now doesn't have the option to play a push out. His only options are to either take the shot on that it's on the table or pass it to player A. If he pass it to player A, player A does have again 2 options: either take the shot on, or play a push out. If he plays a push out, then again player B will see the shot and he can either play or make player A shoot again.