How to solve the system data sqlclient sqlexception 0x80131904 error

Question

I've used C# and SQL Server 2008 R2 to construct an application that runs flawlessly on my PC. What is my connection string?

connectionString="Data Source=KELVIN-PC;Initial Catalog=LMS;User ID=sa;Password=temperament"

However, I have installed it on a another PC and uploaded it. Additionally, I transferred the LMS.bak file to the computer and restored the database on MS SQL Server 2008 R2 there. However, it always generates the following error when attempting to connect to the database on the other machine:

System.Data.SqlClient.SqlException (0x80131904): A network-related or instance-specific error occurred while extablishing 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. (provider: Named pipes Provider, error:40 - Could not open a connection to SQL Server).....

I've spent hours scouring the internet, but the majority of the solutions varied somewhat from mine and some of the advice I followed wasn't actually effective. Can someone please help me with this?