In a clustering, each cluster node or every node that is a candidate for being a cluster node should have two Network Interface Cards (NICs) as a base requirement — one card can be used to configure the Public Network and second can be used to configure the Private Network.
Public Network: In a clustering environment, a network that is exposed to the public, that is, client systems to connect to is called the Public Network.
Private Network: A network that is used to connect two or more cluster nodes is known as the Private Network, also known as the Heartbeat.
SQL Server uses this Heartbeat network for health detection between two or more nodes, especially in a single-node failover cluster. Consider a situation where we have configured a single-node failover cluster — Node1 and Node2. Node1 is holding ownership of the cluster resources and Node2 is in passive mode and has no access to or ownership of any cluster resources.
In case of failure or unmanaged shutdown, or for any reason if Node1 is rebooted or shut down, the Private Network or Heartbeat between these two nodes fails and the Node2 of the cluster will become the active node. It will now have ownership of and access to all the cluster resources until it fails over to Node1.
SQL Server MVP Brad McGehee, in his article, has mentioned some excellent points that should be considered while configuring the Public and Private Networks. The article can be found at http://www.sql-server-performance.com/articles/clustering/cluster_infrastructure_p1.aspx.