Is your instance on Computer A your default instance or is it a named instance?

Do you have a DNS that will solve Computer A name for your Server B?

Do you use the same network or your Computer A is in a different network than your Server B?

If different network, can you connect to your Computer A from Computer A using external IP?

Commonly you would use something like this:

ServerName\InstanceName

ServerName,9999 (ServerName,PortNumber)

192.168.0.5\InstanceName (IPAddress\InstanceName)

192.168.0.5\9999 (IPAddress\PortNumber)

http://www.ServerName.com\InstanceName(URL,InstanceName)

http://www.ServerName.com,9999 (URL,PortNumber)