Node.js is a platform for writing JavaScript applications outside web browsers. This is not the JavaScript we are familiar with in web browsers! For example, there is no DOM built into Node.js, nor any other browser capability.
Beyond its native ability to execute JavaScript, the bundled modules provide capabilities of this sort:
Command-line tools (in shell script style)
An interactive-TTY style of program (REPL which stands for Read-Eval-Print Loop)
Excellent process control functions to oversee child processes
A buffer object to deal with binary data
TCP or UDP sockets with comprehensive event-driven callbacks
DNS lookup
An HTTP and HTTPS client/server layered on top of the TCP library filesystem access
Built-in rudimentary unit testing support through assertions
The network layer of Node.js is low level while being simple to use. For example, the HTTP modules allow you to write an HTTP server (or client) using a few lines of code. This is powerful, but it puts you, the programmer, very close to the protocol requests and makes you implement precisely those HTTP headers that you should return in request responses.
In other words, it's very easy to write an HTTP server in Node.js, but the typical web application developer doesn't need to work at that level of detail. For example, PHP coders assume that Apache is already there, and that they don't have to implement the HTTP server portion of the stack. The Node.js community has developed a wide range of web application frameworks such as Express, allowing developers to quickly configure an HTTP server that provides all of the basics we've come to expect—sessions, cookies, serving static files, logging, and so on—thus letting developers focus on their business logic.
Quit scratching your head already. Of course you're doing it, scratching your head and mumbling to yourself, "What's a browser language doing on the server?". In truth, JavaScript has a long and largely unknown history outside the browser. JavaScript is a programming language, just like any other language, and the better question to ask is "Why should JavaScript remain trapped inside browsers?".
Back in the dawn of the web age, the tools for writing web applications were at a fledgling stage. Some were experimenting with Perl or TCL to write CGI scripts, and the PHP and Java languages had just been developed. Even then, JavaScript saw use on the server side. One early web application server was Netscape's LiveWire server, which used JavaScript. Some versions of Microsoft's ASP used JScript, their version of JavaScript. A more recent server-side JavaScript project is the RingoJS application framework in the Java universe In other words, JavaScript outside the browser is not a new thing, even if it is uncommon.