In a clustered Strategy Intelligence Server 9.x and 10.x environment, the primary node is determined by the order in which nodes are added to one another when the cluster is created. In other words, primary node is the node that the other node joins itself to. For instance, say there are two nodes in a cluster - Node A and Node B. When Node B joins Node A, A becomes the primary node for the cluster of A and B.
If server startup membership is specified in Strategy Intelligence Server Configuration as shown below, then the first node that is up and running will be the primary node, as all other nodes would automatically join this first node.

More information on the above setting can be found in the following Strategy Knowledge Base technical note:
KB13627 (KB5300-802-0774): How to specify which servers can reform a cluster at startup in Strategy Intelligence Server 9.x and newer
The cluster membership can be seen in the Cluster Monitor. The first node listed in the cluster membership is the primary node. For more information on cluster membership in Cluster Monitor, refer to the following Strategy Knowledge Base technical note:
KB13192 (KB5303-8X-0724): How to view the primary node of the cluster for a project in Strategy Intelligence Server 9.x and newer
What happens when the primary node goes down or dies?
In such a situation, the second alive node in the original cluster membership takes over as the new primary node. For instance, say there are three nodes in a cluster - Node A, B and C in that particular order in the cluster membership. When A goes, down, B would take over because it is the second in the line in that cluster membership list.