Node's Require, RequireJS, Browserify and Asynchronous Module Loading

I have issues with Require for a year and I want to clarify the issue besides approving the things I already learned. Other than thousands of questions when I type require.js I'll be happy if someone help to make it clearer for the community too.

• Browsers don't have the require method defined, but Node.js does. So that Browserify can write code that uses require in the same way using Node.
• Node.js does not use Require.js. Require.js was built that we can have asynchronous module loading in the client's browser.

Well, whenever I get an error like below,

After googling/stackoverflowing then I started to use Require.js hence, even it succeeds for some situation then it turned to the error below somehow,

And then with Browserify it definitely succeeds with some cases I tried hence, especially when with asynchronous module loading, it fails like below each single time.

Well it's a bit long but I'm not clear of the main practical usage and I ran into lots of fellow struggling with the issues related this. Thanks for any effort for clarification.

via Erhan Yaşar