Use cURL when you want to grab something from another website.
You can use cURL to help you negotiate your way into a site which may require a POST or a GET submission with say a user/password and even cookies.
Go in, get the response (usually though not always, a html stream) cache the output on your server then analyse the output to extract what you want.
You might also grab the output of an API, or just try and detect the return code, 200, 404 or whatever.
This is the main reason I have used it. Does that help you?