I am receiving this error message on one of our server while publishing a mobile project in mobile project center to ArcGIS server (10.1).
What connection is this one referring to and how to resolve this?
This is the type of error message one can get when one tries to make an HTTP request in an application (ArcGIS Mobile) that is built on the .NET Framework. It is a general Microsoft error message that is not related to the Esri part of the software, i.e., it is the .NET part that generates that message but the cause can come from both sides.
It would be really hard to troubleshoot this issue without getting hands at the software and the environment. I would stronly suggest opening an incident with Esri Support Services since there can be lots of things that can cause this problem (maybe the data piece is too large, you are getting a time-out on a client side; ArcGIS Server has a time-out that doesn't allow you to work with a service that long; OS/network related time-outs or policy settings blocking sending the data; etc).
Some general information from Microsoft:
This problem occurs when the client computer cannot send an HTTP request. The client computer cannot send the HTTP request because the connection has been closed or is unavailable. This problem may occur when the client computer is sending lots of data.
At the same web page you can refer to potential resolutions of the problem, but most of them are applicable only to .NET developers and not relevant for ArcGIS Mobile users which work with already built application.