The google play store lets you log in on your google account on PC, then when you're looking at apps, it will list all the devices you have using that account that are compatible with that app. When you purchase the app, you can specify which device to install it on, and it will do so automatically.. unless you have your options set to not allow it to do so for some reason, but that's not the default. Or if your device currently is on cellular data instead of wifi, then it defaults to not installing until you're back on wifi.
It was all part of the update to "Google Play Services" that came down a few android versions ago. Google can now install/uninstall apps on your phone without user intervention. What a brave new world, eh?