cURL is a very very important PHP library. It is built on the libcurl library which is designed for file transfer over most protocols and is supported on most operating systems. In PHP cURL is designed to add quick easy file transfer functionality to webpages. libcurl and curl are command line functions, the use of cURL is script based but still invokes to command line function of the library.
An example would be invoking cURL to retrieve www.google.com. Instead of a web browser like response that would display Google's front page, this would return the source code for the page.
cURL is usually used in conjunction with HTML parser scripts, regex, and the like, to grab parts of webpages and republish them.