If you have Office 365 and SharePoint online, you have a option to create private site collections and Public Web Sites.
To create Public Web Site using SharePoint Online, just go to SharePoint Online Administration center and click on Site Collections
In Site Collections | Click New | Public Web Site
In create your web site window you need to enter following information
Type a descriptive title for your new site collection. The title will be displayed on each page in the site collection.
To create your Website, select a domain name from the list. If you provided your own website domain on the Microsoft Office 365 website with the domain intent enabled for “SharePoint Online,” you can select the name from the list.
If you choose to use a verified domain name, you must modify your DNS settings as described in the DNS Information dialog box in order to make your site visible on the Internet.
Enter the primary administrator name for your site collection. Only one user name can be provided.
When you have all your information and click Ok to create SharePoint Online Public Site
Now your SharePoint Online Public site is ready.
If you want to customize your SharePoint Online Private Site, open your site using SharePoint designer and start customizing.
See my other blog about how you can connect SharePoint online sites using SharePoint Designer.