From Facebook Developer Wiki

The Base Domain field is an optional field for use with Facebook Connect implementations.

Purpose

You typically specify a base domain if you are trying to run your application on multiple subdomains. For example, chicago.citysearch.com and sanfrancisco.citysearch.com are different subdomains.

Facebook Connect stores user credentials in cookies on your application's domain. By default, a cookie set on chicago.citysearch.com would not be readable on sanfrancisco.citysearch.com - the browser treats them as separate domains. If the base domain is set, then Facebook will intentionally set the cookies on the base domain, thus making the cookies readable across multiple subdomains. This allows you to share one authentication session across multiple subdomains.

Note: There is no way to share a cookie across multiple domains. So, for example, if you have a site cnet.com and news.com, then there is no way to make the browser send the same cookies to both those domains. Each of these would require separate API key and separate authentication.

For more information, see Supporting Subdomains In Facebook Connect.

Specifying the Base Domain

You can specify a base domain by editing your implementation's settings on the Connect tab in the Facebook Developer application.

Alternately, you can set it programmatically using admin.setAppProperties.