GWT normally only shows about 20% -30% of the backlinks that Google has actually found. In many cases Googlebot will discover a new backlink to your site, but GWT will never show it. Evidently Google with-holds full information on backlinks as a way to try to thwart artificial link-building activities by webmasters.

As for how long it takes Googlebot to discover a new backlink, that depends on the crawl rate for the webpage where the backlink is located. If Googlebot crawls the page everyday, it will find the new backlink within 24 hours of its creation. But if Googlebot only crawls the page once a month, then it could take up to 30 days for it to discover the new backlink.

Let's say google has discovered all my backlinks but why wouldn't my ranking have changed after 3 months ? does it take another 6 months after the backlinks were discovered to change the ranking of my site ?

There are many, many possibilities for an answer. The basic idea is that not all backlinks are created equal, and some give only give a very small amount of target power to their target page. This is not a complete list, but here are at least some of the differences:

1. Page Rank of the linking page
2. How many other links on the linking page
3. How close to the top of the content the link appears
4. Position of the link on the page template altogether - in the footer barely offers anything, for example, but high in the main content area can offer a lot more.
5. Penalties and filter against the linking page or site
6. Penalties and filters against the target page or site

I am confused with these backlinks. The idea is for another site to link to my blog? is that right, and site A would imbed links to site B within site A?

If so, does the page have to in a blog, is thqat preferable to a site?

No there's no reason a backlink has to be in a blog to give the target site a ranking boost. The idea is that Google will value a page more strongly if other websites discuss it and link to it of their own free will - that was the concept behind the invention of PageRank.

A blog IS a kind of site, one with a certain type of content management system. That alone doesn't mean anything about the value of its outbound links.

well how do you find the other websites to link to your website?

Backlink data is available in Webmaster Tools. A limited amount is also available with the link: operator. In addition there are paid services such as SEOMoz and MajesticSEO that do their own spidering and sell you the data they find for a fee.

Just to add to tedster's list:

7. what anchor text was used in the link to your site
8. Whether the links have a nofollow tag (which is common for links that you would get when posting a comment in a blog's comment section)