Javascript library uses require() but I don't have or use nodeJS?

Libraries should ideally support the following, depending on its environment. Lets say you are using a library called "MyLib.js".

No module loader detected

window.MyLib

Requirejs detected

define(['MyLib'], function (MyLib) {
  // Do something
  return {};
});

CommonJS detected, like node or use of browserify or bower

var MyLib = require('MyLib');

Not all libs conform to this, but they should. Maybe the lib you were looking at only supports Node. Looking at the source of jQuery you see something like this:

if ( typeof module === "object" && typeof module.exports === "object" ) {
    // For CommonJS and CommonJS-like environments where a proper window is present,
    // execute the factory and get jQuery
    // For environments that do not inherently posses a window with a document
    // (such as Node.js), expose a jQuery-making factory as module.exports
    // This accentuates the need for the creation of a real window
    // e.g. var jQuery = require("jquery")(window);
    // See ticket #14549 for more info
    module.exports = global.document ?
        factory( global, true ) :
        function( w ) {
            if ( !w.document ) {
                throw new Error( "jQuery requires a window with a document" );
            }
            return factory( w );
        };
} else {
    factory( global );
}

Node.js is now used a lot to manage JavaScript projects, even if the project is client side. For example, Grunt, Bower, Browserify, Gulp and plenty other build tools run on Node.js, even though you can use them on client-side projects. Using those tools doesn't make your project depend on Node in production. Node is only used for development. To install those tools, one uses npm, which is a package manager. Like Maven or Ivy, npm will install packages and their dependencies by downloading them from the internet.

Libraries which install instructions involve npm are meant to be used in Node, but may be used in the browser after they have been converted with Browserify. Download the library with npm, then convert it to browser style using Browserify (which you install using npm, because itself runs on Node). You should get a single JavaScript file that you can import in your client-side project.

Libraries that are especially targeted for the browser will often refer to Bower as the install method instead of npm. Bower is also a package manager, but it is designed to download and install library written for browsers, not Node. If a library you want is available on Bower, you are able to download it and all its dependencies with bower install <lib>. Bower will put all files in a bower_components folder in the current directory. You can then copy those files to your project, or make your project import them directly from this folder.

So, browserify is simply a tool which makes it possible to use node-style modules in the browser. Yes, you do need to have node.js installed in order to use npm and browserify. But these times you need node.js for the most of your frontend toolsets.

npm is full of modules that are written in JavaScript and also run in the browser. With browserify you can use these modules in the browser.

It works by shimming the whole require mechanism and making it work in the browser. This also means that you can organize your code in modules:

// add.js
module.exports = function(x, y) {
    return x + y;
}

// app.js
var add = require('./add.js');
var result = add(7, 8);

Now you could generate your bundle (the only script you need to include in your html) by just running browserify app.js -o bundle.js.

If you don't like the browserify approach you can also use the --standalone option to generate a JavaScript file in the UMD format. You could then simply include this in your html and use it with window.add for the previous example.

The require statement is handled by require.js, which can be used standalone in a javascript environment. It is a module loader which optimises loading dependencies in the browser. There is a Node.js implementation for it, but that doesn't mean that you have to use Node, you can just include require in your project.

(p.s: npm is Node Package Manager, and would be unnecessary for your project unless you're using Node. It streamlines the inclusion of javascript node modules into your project.)