Here's another chaining example on using q.js. This doesn't have any error handling, as I just want to demonstrate the chaining concept. Please read the comments carefully, as I start out with a non-q example, to show the order of flow. Please post comments if there's anything that isn't clear and I'll try to revise it as needed.

q_example.js

// Q sample by Jeff Cogswell

/*===========
 We want to call these three functions in sequence, one after the other:
 
 First we want to call one, which initiates an ajax call. Once that 
 ajax call is complete, we want to call two. Once two's ajax call is 
 complete, we want to call three.
 
 BUT, we don't want to just call our three functions in sequence, as this quick 
 demo will show. Look at this sample function and think about what order 
 the console.log calls will happen:
===========*/

function demo() {
    $.ajax( {
        url: '/',
        success: function() {
            console.log('AJAX FINISHED');
        }
    });
}

console.log('Calling demo');
demo();
console.log('Finished calling demo');

/*====
The function returns almost immediately, before the ajax call is complete.
That means we will likely see 'Finished calling demo' before we see the 
results of the ajax call: 
====*/

//Calling demo
//Finished calling demo
//AJAX FINISHED

/*==== 
If we want to chain a following function, when do we call it? 
We call it from inside the success function:
====*/

function demo() {
    $.ajax( {
        url: '/',
        success: function() {
            console.log('AJAX FINISHED');
            
            // >>>> THIS IS WHEN you would call another function <<<<<
            
        }
    });
}

/* ==============
 Now let's try using q.
=============*/


function one() {
    
    var deferred = Q.defer(); // Don't worry yet what this is
                              // until after you understand the flow
    
    console.log("Starting one's ajax");
    $.ajax( {
        url: '/',
        success: function() {
            
            // Here's where you want to call the next function in the
	    // list if there is one. To do it, call deferred.resolve()
            console.log('Finished with one. Ready to call next.');
            deferred.resolve();
            
        }
        
    });
    
    // The deferred object has a "promise" member, 
    // which has a "then" function
    return deferred.promise;
}

function two() {
    var deferred = Q.defer();
    console.log("Starting two's ajax");
    $.ajax( {
        url: '/',
        success: function() {
            
            // Again, this is where you want to call the next function
	    // in the list if there is one.
            console.log('Finished with two. Ready to call next.');
            deferred.resolve();
            
        }
        
    });
    // The deferred object has a "promise" member,
    // which has a "then" function
    return deferred.promise;
}

function three() {
    var deferred = Q.defer();
    console.log("Starting three's ajax");
    $.ajax( {
        url: '/',
        success: function() {
            
            // Again, this is where you want to call the next function
	    // in the list if there is one.
            console.log('Finished with three. Ready to call next if there is one.');
            deferred.resolve();
            
        }
        
    });
    // The deferred object has a "promise" member, which has a "then" function
    return deferred.promise;
}

// Test it out. Call the first. Pass the functions 
// (without calling them, so no parentheses) into the then calls.

one()
    .then(two)
    .then(three);

/* =====
Think about where the "then" function comes from. Each function 
creates a new defer instance and returns that object's promise 
member. That promise object has a "then" function. On return 
from the first function, you get back a defer function, and 
call the "then" function, passing the *next* function that is 
to be called. Internally, Q stores that function. When your 
ajax call returns, in your "success" function, you call the 
next function by calling deferred.resolve().
======*/

Even this example is not clear. For one thing, what is the $ variable? Is it the jQuery handle? If so, where is the require statement for jQuery? Is this only for the browser? Where is the script tag to load jQuery or Q? I tried running it in NodeJS (since that's what I want Q for). Is there a $.ajax function for the server side?

Wouldn't it be much clearer/respresentative to rewrite the example as

one()
.then(new function() {
    return two();
})
.then(new function() {
    return delay(2000);
})
.then(new function() {
    return three();
})
.fail(new function(err) {
  console.log("Something went terribly wrong between 1, 2 and 3! " + err);
});

Thanks for the great example! Here's a JSFiddle for experimenting: https://jsfiddle.net/josepi08/0nevk0a9/

New to Q and it's Promise API's,
one()
.then(two)
.then(three);
.done // Looking at the doc's https://github.com/kriskowal/q/wiki/API-Reference should you call "done" at the end?