The error message is pretty explicit!
A network-related or instance-specific error occurred while establishing a connection to SQL Server. The Server was not found or was not accessible. Verify that the Instance name is correct and that SQL Server is configured to allow remote connections.
Look at your code, and find the connection string you are using to connect to SQL Server. If you hard coded it - that's a problem, and you need to move it to a configuration file immediately.
When it's in a config file, you will need to edit it to reflect the path to the SQL server instance holding your database as seen by the new PC. Probably, it is set to your development machine incstance "LOCALHOST\SQLEXPRESS" or similar, with windows authorization set on. This needs to be changed to reflect the SQL instance that the client PC will be using, complete with the correct login details.
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