After I did a clean installation of MS SQL Server, I created a new user using SQL Server Authentication. When I used MS SQL Server Management Studio to connect, I got the "Login failed for user ..." error:
I double-check my password and give it admin roles. Everything seems to be fine but the login still failed.
I found out that by default, MS SQL server is set to Windows Authentication mode. So, if you try to access using SQL Server Authentication, it will not work. But, why allow that option in the first place to connect.
Anyway, from the database server Properties > Security, under the Server authentication section, I select SQL Server and Windows Authentication mode and then restart MS SQL server.