What would you use for full fledge javascript applications like New Twitter, Gmail, etc.
jQuery is still just fine for large-scale applications. You can organize the code any way you have to. Its just Javascript after all.
If your app gets so large that multiple people are doing Javascript though, what you need to do is work on what is considered the boring stuff like version control policies, build processes (look at stuff like Sprockets (http://getsprockets.org/) for some inspiration for instance.
Other than that, jQuery is likely to be a fantastic choice in virtually any front-end engineering that you do.
Forgive my elementary level question. I'm not a programmer, I'm just trying to learn about what my contracted coder is doing.
So is it better to write naked javascript or use any other framework?
I read some where people using YUI, Dojo Toolkit, etc.
Naked Javascript can exist alongside jQuery, and jQuery can exist alongside other frameworks as well, so if there is something that jQuery will make easier, use it. If there are things that plan Javascript can do just fine, use that. Choose the best tools for the job, I say.