When to choose Windows authentication?
If you directly logged in to the remote machine then use this type of authentication. It will allow you to connect using the same username of remote machine and you do not need to enter SQL database username and password. If you are logged in Remote machine as a administrator then you will have full admin rights of all databases of server.
When to choose SQL server authentication?
If you are connecting to your database from a remote location, then you need to use SQL Server Authentication. You will have to specify database Server Name or IP Address, port number and also Database Username and Password to authenticate.
If you face any trouble connecting to your Database Server then please be sure to check your local Firewall to ensure that port 1433 or 1533 is not blocked there. The port number detail depends on your hosting provider which they have enabled. Please contact your hosting provider to get exact detail.