Elasticsearch: Significance of Node_name and Cluster_name

How do I setup different nodes(Network computers) to belong to same cluster?

For example: 

       Cluster Name:    my_clustername

        I've three nodes(network computers) where I want to store data
        and   10.65.XXX.X5
        with 15 shard and 3 replicas.

I want all three computers to store data for single cluster (my_clustername)

So how can I tell Elasticsearch to store data on these nodes so that they discover
and auto-join each other?

Do they(Nodes) needs to have same name?

Make sure ES binds to the correct network(s) and not just localhost (the default) by adjusting the network.host setting as needed. Then configure unicast discovery so that the nodes know about each other. They need to have the same cluster name but shouldn't have the same name (not sure if it's forbidden to have two nodes with the same name but it's definitely a bad idea).