I have been building IMO a really cool RIA. But its now close to completion and I need to test it to see if there are any bugs or counter-intuitive parts or anything like that. But how? Anytime I ask someone to try to break it, they look at it for like 3 minutes and say "it's solid". How do you guys test things? I have never used a UnitTest before, actually about 3 months ago I never even heard of a unit-test, and I still don't really understand what it is. Would I have to build a whole new application to run every function? That would take forever, plus some functions may only produce errors in certain situations, so I do not understand unit tests.
It's based on Xcode and Cocoa development, but this video is still a great introduction to unit testing. Unit testing is really something that should be done alongside development, so if your application is almost finished it's going to take a while to implement.
First thing is to systematically make sure everything works in the manner you expect it to. Then you want to try it against every realistic hardware with software installed combination that is feasible and appropriate. Then you want to take every point of human interaction and try putting as much data in, no data in, and special data that may cause exceptions. The try doing things in an order or workflow you did not expect sometimes certain actions depend on others. You and your friends will naturally do those steps in order, what happens when someone doesn't? Also, having complete novices use it is a good way to see odd things users might try.
Firebug has a good profiler for web apps. As for testing JS files, I use Scriptaculous. Whatever backend you are using needs to be fully tested too.
But before you do that, you need to understand what unit testing is. Unit testing is verifying that all of the individual units of source code function as they are intended. This means that you verify the output of all of your functions/methods. Basically, read this. There are different testing strategies beyond unit testing such as integration testing, which is testing that different modules integrate with one another. What you are asking people to do is Acceptance testing, which is verifying that it looks and behaves according to the original plan. Here is more on various testing strategies.
PS: always test boundary conditions
The question is pretty open-ended so this post won't answer all your question. If you can refine what you are looking for, that would help.
There are two major pieces of testing you likely want to do. The first is unit testing and the second is what might be called acceptance testing.
Unit testing is trying each of the classes/methods in relative isolation and making sure they work. You can use something like jUnit, nUnit, etc. as a framework to hold your tests. Take a method and look at what the different inputs it might expect and what its outcome is. Then write a test case for each of these input/output pairs. This will tell you that most of the parts work as intended.
Acceptance testing (or end-to-end testing as it is sometimes called) is running the whole system and making sure it works. Come up with a list of scenarios you expect users to do. Now systematically try them all. Try variations of them. Do they work? If so, you are likely ready to roll it out to at least a limited audience.
Also, check out How to Break Software by James Whittaker. It's one of the better testing books and is a short read.