Cannot connect to the LobSystem (External System).
Sometimes the message gets lost in translation. A colloquialism finds its way into the names of XML elements and error messages. What is a lobsystem? And what has it got to do with databases and external content types? I'm just visualizing a lobster. Then again, that may be because it's nearly lunchtime and I'm getting hungry.
So, presumably, the author of this message was thinking of a LOB System, where LOB is an acronym for "Line of Business". And some databases and back-end systems contain information connected with a particular business line, if the back-end system contains business data, and if the system is a business system at all. So the notion of an LOB System becomes somehow synonymous with any kind of back-end system that you are talking to using BCS. So it means you can't talk to the database. Aha! Thank you.
And the most common reason for not being able to get to the database, apart from someone forgetting to plug it into the network or switch it on, is permissions. And usually the database is SQL Server and we don't have permissions set up in SQL Server. Remember that the user account that needs access to the database will depend on the method used to authenticate. If you use pass-through or impersonation using Secure Store Service (individual) it will be the user's identity. If you use RevertToSelf it will be the BDC identity, which means the application pool account. If you are impersonating using a group account it will be the account you set up in the Secure Store Service when you set up the application.
You can quickly check for problems by going to the database server and looking in the application event log for the tell-tale SQL Server authentication errors. For a good article about dealing with this problem see this blog post.