I built a quiz taker app with Ruby on Rails, using the MVC framework. I want to do basically the same thing with pure JavaScript, and add more features with jQuery for mobile devices and special effects, since jQuery is just awesome like that.
I have looked around on sites like TodoMVC for comparisons on frameworks, but I'm foreign to how these frameworks function. Why do they use Collection rather than Controller in their MVC definition? It seems to me that these are just client-side frameworks. Would I use something like Backbone.js for client-side work and Node.js for server-side?
I'm just unsure as to how development with pure JavaScript functions. I need to create a database to store quiz and user information and be able to access that database when viewing most pages, so I feel the MVC framework is the best way to go.
Any suggestions as to where to start?
If you are looking for a JavaScript MVC-like framework take a look at Express.js http://expressjs.com
Express.js is more like Sinatra than Rails, but it will give you a good foundation on the server side.
Wow, that's quite a broad post; let's take things one at a time
Why do they use Collection rather than Controller in their MVC definition?
The exact answer varies by library, but a simplistic answer is that the Controllers in most Javascript apps doesn't really need any framework; they can be raw Javascript and work just fine as they're usually not very complex.
Furthermore, parts what you might consider to be "the Controller" are often provided separately. The primary example of this is Backbone's Router object: it is similar to the implied routing (and routes.rb) in Rails, or the urls.py in Django. Just as those frameworks don't consider routing to be part of "the Controller", Backbone (and similar frameworks) provide routing as a separate piece ... even though it could fall under the C in MVC.
Similarly, much of the DOM manipulation functionality that jQuery provides would normally belong in the Controller of an MVC app, so in a sense jQuery helps you build your controllers; it's just not explicit the way Backbone.Model helps you build your models.
Would I use something like Backbone.js for client-side work and Node.js for server-side?
That's really apples and oranges; to put in server-side terms, Backbone is more like Rails, and Node is more like Ruby (or Mongrel or something). So yeah one's client-side and the other's server-side, but the differences are deeper than that.
Any suggestions as to where to start?
Pick a framework and get your hands dirty! Seriously, you can spend hours reading reviews of the different frameworks, and still be no closer to making a decision at the end than when you started (I speak from experience). But if you just pick one and try it, you'll likely find that it either "gels" with you or doesn't pretty fast.
Personally, I'd recommend starting with Backbone just because A) it's very popular these days, and B) I'm biased: I use it on a daily basis and love it. Also, it probably would fit you well because it was created by the CoffeeScript guy (and CoffeeScript was his attempt to make Javascript more Ruby-like). However, Ember.js is also very popular these days, and Hector mentioned ExpressJS, which I know nothing about but could be cool.
But the point is, pick one, do a hello world, or maybe something a little more complex like an introductory tutorial (Backbone has one around a To Do app). A few hours of coding with it will tell you far more than I (or anyone else on Stack Overflow) ever could about whether it's right for you.
I would definitely look at Google Closure. I have started to use it with LimeJs on a personnal pet project and really got caught with it's compiler and it's modular design.
It's not really MVC, but since JavaScript should be thought more in an asynchronous fashion, I tend to think more of it as event driven than simply a request-pipeline-response way.
So, Closure Library and jQuery / jQuery UI (both on Google API) for the client side and ExpressJs for the server side. Also, take a look at this framework comparison chart, you might find it interesting.