Home Assistant is a python application to automate your house. It is probably best installed on linux, but apparently also works on windows and OSX.
Hassio is a dockerised version of home assistant, plus a dockerised supervisor. It can be installed on any linux, I am not sure about OSX and windows. It also has the facility to to run hassio addons, which are separate applications like a dhcp server, a mosquitto server - pretty well anything you can put in docker can be made into an addon.
Hassos is a stripped down linux operating system designed just to run hassio (plus those addons of course).
To complicate matters further, you can also run home assistant virtual machines like virtualbox. This can be done a number of ways:
Run a VM and install a linux OS, then install ha as you would in a computer running linux as it’s host OS.
Run a VM and install a linux OS, then install hassio as you would on a linux box.
Run a VM and install hassos as a VM.
I wouldn’t recommend the VM route, unless you really absolutely only have one computer in your house and you need it to run windows as the host OS.