var express = require('express');
var app = express();
This is how we create an express application. But what is this 'express()'? Is it a method or a constructor? Where does it come from??
Is it a method or a constructor?
Neither; it's a function, although if you said "method" I don't think anyone would give you a hard time.
A method is a function attached to an object. In JavaScript, methods are just mostly functions that you reference via object properties. (Update: As of ES2015, if you use method syntax to create them, they're slightly more than that because they have access to super.)
A constructor, in JavaScript, is a function you call via the new operator. Even though other functions may create things, we don't typically call them "constructors" to avoid confusion. Sometimes they may be "creator" or "builder" functions.
Where does it come from?
ExpressJS is a NodeJS module; express is the name of the module, and also the name we typically give to the variable we use to refer to its main function in code such as what you quoted. NodeJS provides the require function, whose job is to load modules and give you access to their exports. (You don't have to call the variable express, you can do var foo = require('express'); and use foo instead, but convention is that you'd use the module's name, or if only using one part of a module, to use the name of that part as defined by the module's documentation.)
The default export of express is a bit unusual in that it's a function that also has properties on it that are also functions (methods). That's perfectly valid in JavaScript,¹ but fairly unusual in some other languages. That's why you can create an Application object via express(), but also use express.static(/*...*/) to set up serving static files.
¹ In fact, it's completely normal. Functions have a couple of standard methods by default: call, apply, and toString for instance.