I haven't used Hadoop, but I understand that it relies on Unix shell commands, so if you are on a Windows machine, then you either need a Unix-shell for your Windows machine (which is what Cygwin does), or you need a Linux virtual machine running inside your Windows machine (which is what VMWare does). Alternatively, you can either install Linux to dual-boot on your current machine before installing Hadoop, or find a machine that already has Linux on it and install Hadoop there instead.
I don't use Cygwin, so I don't know how you would install/run Hadoop with Cygwin, but I'm guessing you would open a Cygwin shell and then use Unix-style commands as indicated in the
Hadoop instructions.
VirtualBox is an easy free alternative to VMWare - you can install VirtualBox on a Windows PC, then install a Linux VM to run inside VirtualBox e.g. here is a
quick guide to installing Ubuntu Linux on VirtualBox. Once you've got Linux running inside VirtualBox, I guess you would log into your Linux VM and then follow the instructions for installing/running Hadoop on Linux.
If you're using Ubuntu, remember you may need to use "sudo" for running some commands that require extra permissions e.g. "sudo apt-get install ssh". Also, running a VM inside Windows takes extra memory, so make sure you have enough RAM to run both OS at the same time. If you dual-boot Linux instead, obviously this is not such a problem as you are either running Windows or Linux but not both at the same time.
FWIW, it may be slightly more work to install Linux (as a VM or dual boot) on your Windows PC, but once you've done this it's often much easier to get/install/use the Linux versions of many open source tools than the equivalent Windows versions.