The technical explanation is that it's the realm name sent back to the browser when it notifies it that the page requires Basic authentication via a 401 response code.
For example the HTTP header for such a response would contain:
WWW-Authenticate: Basic realm="my realm-name"
The browser if it hasn't already asked for a username and password will prompt the user to enter one for "my realm-name".
Any subsequent pages that generate a 401 with the same realm will use the username and password already entered by the user, so will NOT require the user to log in again.
In theory you could give two different applications hosted by the same server, the same realm-name, and therefore only require the user to login once. A realm is a way of grouping pages / urls together with the same authorisation requirements.
In reality the realm-name is just displayed as a descriptive line of the browser's login prompt.
For more information on realms and basic authentication you can read the HTTP specification:
http://www.w3.org/Protocols/HTTP/1.0/spec.html#AA Rufus.