Googlebot is a web crawling bot owned by Google. Its purpose is to collect all the different URLs from the websites. It is also sometimes called a "spider".

I have written the following code inside my robots.txt file:

User-agent: *

Disallow:

User-agent: googlebot

Allow: /

Is my robots.txt is correct?

Crawling is done by search engine bots (eg. Googlebot) to check new and updated pages to rank them. It is analyzing webpage links and content to rank according to that.

I would go to http://www.google.com/addurl then enter your website's home page URL, e.g. http://www.yoursite.com/. As Google states, you only need to submit your domain URL to them and Googlebot will be able to find the rest of your web pages.