Thought I'd write exactly what I did (Windows 10, 64-bit):
From the download page https://curl.haxx.se/download.html choose the download wizard https://curl.haxx.se/dlwiz/
Choose curl executable.
Choose Win64.
Choose generic.
Choose any.
Choose x86_64.
Choose the first recommended option. For me this was:
curl version: 7.53.1 - SSL enabled SSH enabled. Provided by: Viktor Szakáts. This package is type curl executable You will get a pre-built 'curl' binary from this link (or in some cases, by using the information that is provided at the page this link takes you). You may or may not get 'libcurl' installed as a shared library/DLL.
The file is packaged using 7zip. 7zip is a file archiving format.
Click download.
You should have the file curl-7.53.1-win64-mingw.7z in your downloads folder.
Install 7-Zip if you don't have it.
Right-click, 7-Zip, Extract Here. Copy and paste the extracted file somewhere like Z:\Tools\
If you look in the bin folder you'll see curl.exe. If you double-click it a window will quickly flash up and vanish. To run it you need to use the Command Prompt. Navigate to the bin folder and type curl followed by your parameters to make a request. You must use double-quotes. Single quotes won't work with curl on Windows.
Now you'll want to add curl to a user's Path variable so you don't have to navigate to the right folder to run the program. Go to This PC, Computer, System Properties, Advanced system settings, authenticate as an administrator (you're not running as admin, right? Right?) Environment Variables, System variables, look at the list and select Path, then Edit, then New, then, e.g.
Z:\Tools\curl-7.53.1-win64-mingw\bin
You can add a trailing backslash if you like, I don't think it matters. Click move up until it's at the top of the list, then you can see it easily from the previous screen. Click OK, OK, OK, then crack open a Command Prompt and you can run curl by typing curl from any folder, as any user. Don't forget your double-quotes.
This is the answer I wish I'd had.
C:\Windows\System32
directly! See my answer below. – VonC