In this case backup server means mail exchange server or gateway.
its what receives the incoming emails then forwards it to the mail server.
Its where your MX records point.
A backup server is just that, it holds the incoming emails while the mail server is not available then deliver them to the mail server when it responds.
A fail over server is a server that runs in parallel with your primary server.
Both use the same shared storage. If the primary goes down the fail over server takes over.
It seems you are asking about the gateway option.
I have never used the free smartermail option for this but Im think that will work since no email accounts are stored on the server, it just handles incoming email (can handle outgoing too).
The MX records controls where external emails are sent so in your case thats the same as your main mail server.
If the mail server is down, email will get bounced.
If you have a gateway then the MX records will point to the gateways. You should have at least 2 gateways.
So both gateways needs to be down for emails to get rejected.
Where does your MX records point?
both the primary and what you call the backup server?
If they are all on the same internet connection and it goes down everything will not work of course.