A few things to set up on the SQLEXPRESS before client can connect to it remotely.
Go to SQL Server Configuration Manager > Network Configuration > Protocols for SQLEXPRESS and enable TCP/IP. The default installation of SQLEXPRESS does not enable TCP/IP. If you are connecting via other protocols, enable those (obviously).
Go to the host machine’s services.msc and enable the SQL Server Browser service. You may want to change the startup to Automatic as well.
Additionally, you may need to configure SQL Express to use Mixed Mode Authentication. This is normally a one-time configuration during the installation of the server, however, it can be changed manually through the registry at
HKLM\Software\Microsoft\Microsoft SQL Server\MSSQL.1\MSSQLServer and set
LoginMode to 2 (decimal)
Then restart SQL Server (SQLEXPRESS) in services.msc to make it use the new settings.
If users are required, use Microsoft SQL Server Management Studio to add users that can log in with SQL authentication.