Is it not a Javascript engine hosted by the browser on the client machine itself?
No, it isn't.
Server generally has two meanings:
A Node.JS server can be either of those.
In web programming, a Node.JS server takes the place of Perl, Python, Ruby, PHP, Scala, etc. (And like those other languages, Node.JS lets you use JavaScript for non-server and non-web purposes).
Generally the server itself is run directly from Node (e.g. with this library) rather than being embedded in another server like Apache (as is most common for PHP).
A browser doesn't need to be involved at all. If one is, then it will probably be one acting as a client and making a request to the server. That said, tools like PhantomJS can allow a browser to be driven from Node (and other programming languages).