Nostr is the network. It is the same for all. There are different clients (apps), that all connect to the nostr network.

There are nodes. Nodes save what is in the netwotk posted. So nodes save your messages. Mostly your mesages will be posted and saved on different nodes. The admins of nodes decide who and what can be posted on them. Some nodez offere it as a service and you can pay them. You can also set up a node only for you, so your mezsages are allways saved.