Once I've posted a thread or a reply, I could search my post by entering its content on the google.

Why?

cos your post is stored on the internet, and google searches the internet.

Google as well as other competitors like Yahoo use bots to scour the web for information. They check forums like this one too. All text information is saved on their servers... yes this is ALOT of information, terabytes and terabytes, and stays there pretty much forever. It gets cached. You can read about new privacy issues that have arisen because of this:

 

http://www.reuters.co.uk/newsArticle.jhtml?type=reutersEdgeNews&storyID=746393

Which option shall I set in order to let my web site become a part of the google?

No need to set up an option... you can do one of two things... let google find you or submit your site to google. You can let google find you by posting a link to your site on a forum that is already indexed in google's cache system and wait for the bots to find you (more reliable than submission alone). Here is how you can get it indexed: http://www.google.com/webmasters/

 

By the way you can create a meta-tag so that you can put your own description of the site in hte cache (like the title and a small description).

 

Here is how you can make a meta-tag: http://searchenginewatch.com/webmasters/article.php/2167931

 

be sure to make the meta-tag fit with your site otherwise you website will have a lower google ranking (place it comes up when search is entered).

 

Although there is some code that can prevent a bot from idexing you but I doubt that you have that in there.