How do I install cURL on cygwin?

In order to use the command-line version of curl, you need the curl executable. So, run the Cygwins Setup.exe, and select curl (under Net->curl). That one uses libcurl3, which is located in Libs->libcurl3. But libcurl3 will be pulled in as a dependency if it's not already installed. So, just select Net->curl and you're good to go.

If you don't see a certain package, you can access to a full list of ports (also unnoficials, the packages you see on the web) launching the setup.exe with -k argument with value http://cygwinports.org/ports.gpg (example: C:\cygwin\setup\setup-x86.exe -K http://cygwinports.org/ports.gpg).

Doing so, you can choose a lot of extra packages, also extra versions of cURL (compat one). I do that to get Apache, cUrl, php5, php5-curl and some others :)

I don't know if apt-cyg can get those extra packages.