I want to create a Sharepoint site and want to publish an Access web database on it. I want to know is there a way to give different permissions to Sharepoint site users? Some users have just read permissions, some have just add permissions and some have edit permissions.
Provided that you're talking about Access Services service application; yes, you can manage permissions using the default SharePoint mechanism. See http://www.mattjimison.com/blog/wp-content/uploads/2011/09/Access-Services.pdf for a pretty good overview of features.