Download and run Minecraft.msi. This will install Minecraft and create a shortcut in your start menu. If you'd like a version without an installer, you may use Minecraft.exe instead. You do not need Java installed to run either of these.
Minecraft for Windows (alternative)
If you have issues with the new Windows download, you can try the old one by downloading Minecraft.exe. If given the option, save the file instead of running it. Save the file anywhere you'd like, then double-click the program to run. This does require Java to be installed, however.