If you want to publish your site that created by Sharepoint 2010 it is a good solution to allow anonymous access to it.
Let me show how I am doing: (later on I will attach figures too)
1. Open the Central Administration page and Application Management
2. Choose the Manage Web Application
3. Highlight your portal and select the Web Application tab on the ribbon above
4. Choose the Authentication Providers icon
5. On the pop-up window click the Default value
6. At the Anonymous Access option enable the anonymous access and press ok
7. On the ribbon above choose the Anonymous Policy (be sure whether your selection is still active)
8. Be sure that in All Zone the AnonymousUser Policy set to None – No Policy (Save It.)
9. Open your site and choose the Site Settings then choose the Site Permissions
10. Under the Permission Tools choose Anonymous Access icon
11. Check that Anonymous User Can Access: Entire Web Site radio button is selected (if do so, click ok)
12. Check whether you can see the anonymous user as permitted user to the site.