How can I save opened application settings in Mac OS X? In Linux there is an option to save the currently opened windows such that when you shutdown and boot later, these applications are there the way you left them. It is similar to Hibernate in Windows.

Is there something similar in Mac OS X?
You are looking at two different things.

1) Shutdown and boot - get your same apps as per Linux option - see @Am1rr3zA

2) Hibernate - OSX does this already. Just choose sleep from the apple menu
For more details see Macworld explanation and Apple docs