Whatever username you are using on the "remote" machine must be set up on the "server" machine.
So, if you are logged-in to "remote" as "John," there must be a "John" user on the machine that is hosting SQL Server 2008 Express, and unless you are going to provide different credentials, it should have the same password, too.
Connect to SQL Server using Windows authentication. Run sp_readerrorlog and see the complete message which would have come in errorlog when you tried connecting with SQL Authentication. This message is not of much help. You can update the question with message
From error log: Login failed for user 'sa'. Reason: An attempt to login using SQL authentication failed. Server is configured for Windows authentication only.
What is output of this query.
select serverproperty('IsIntegratedSecurityOnly')
It seems like authentication mode is Windows only. If above query returns 1. It means authentication mode is windows go to server . Right click on SQL Server select property and then click on security. Change authentication it to mixed mode by selecting radio button SQL Server and windows authentication mode and restart SQL Server service again.
Best Answer
Ok found the answer after an hour of trying various options.
Server properties - Security (see image)
Shame there is no warning message somewhere when you create SQL Authentication users that they have no access.