Occasionally, and for reasons that are unclear, the CRM Email Router file Microsoft.Crm.Tools.EmailAgent.SystemState.xml gets corrupted which causes the Email Router service to stop (and your email along with it).
If you find your Email Router is not running and a restart does not work-it immediately starts and stops, check the Windows Application event log to see if you see any errors in the MSCRMEmail category.
They actual error can vary, but it contains a message stating some type of problem with the Microsoft.Crm.Tools.EmailAgent.SystemState.xml file.
Here is a sample Microsoft KB article outlining one such error:
Error event when you try to start the Microsoft Dynamics CRM E-mail Router Service in Microsoft Dynamics CRM: "The E-mail Router service could not run the service main background thread"
Regardless of the error, the solution is to delete the Microsoft.Crm.Tools.EmailAgent.SystemState.xml file then restart the Email Router service.