A **private remote node** doesn't run on your local machine, but you have full control over it. A classic example is a node running on a VPS or a server. That node is not on your local machine, but you have full control over it. You can even decide to make it become an open node, allowing people to connect to it.
An **Open remote node** can be used by people who, for their own reasons (usually because of hardware requirements, disk space, or technical abilities), cannot/don't want to run their own node and prefer to relay on one publicly available on the Monero network. Open remote nodes are often simply referred as "remote nodes".