The engine in the curl command-line tool is libcurl. libcurl is also the engine in thousands of tools, services and applications out there today, performing their Internet data transfers.
libcurl is a library of functions that are provided with a C API, for applications written in C. You can easily use it from C++ too, with only a few considerations (see libcurl for C++ programmers). For other languages, there exist "bindings" that work as intermediate layers between libcurl the library and corresponding functions for the particular language you like.
We have designed libcurl to be transfer oriented usually without forcing users to be protocol experts or in fact know much at all about networking or the protocols involved. You setup a transfer with as many details and specific information as you can and want, and then you tell libcurl to perform that transfer.
That said, networking and protocols are areas with lots of pitfalls and special cases so the more you know about these things, the more you will be able to understand about libcurl's options and ways of working. Not to mention, such knowledge is invaluable when you are debugging and need to understand what to do next when things do not go as you intended.
The most basic libcurl using application can be as small as just a couple of lines of code, but most applications will, of course, need more code than that.
libcurl generally does the simple and basic transfer by default, and if you want to add more advanced features, you add that by setting the correct options. For example, libcurl does not support HTTP cookies by default but it does once you tell it.
This makes libcurl's behaviors easier to guess and depend on, and also it makes it easier to maintain old behavior and add new features. Only applications that actually ask for and use the new features will get that behavior.