The cause of this error is that a login is being attempted to a database with the "ApplicationPoolIdentity" account. In a development environment, you would prefer using your credentials to log into a SQL Express database. This can be fixed by doing the following:
Go to IIS Manager
Click on Application Pools
Click on the offending application pool
Click on the "Advanced Settings ..." link on the right-hand-side
Find the "Identity" property and change it from "ApplicationPoolIdentity" to "LocalSystem"