A portable 2.5" USB HDD enclosure works exactly in the same way as a USB flash drive, except you have much larger capacity and can reuse your laptop's old HDD.
Running Windows off entirely off a USB drive means that particular HDD has to stick with one system only, for configuration issues. If you want to run one OS with multiple computers by taking the USB around then you would have to look at a flavor Linux.
If you just want to boot OS installation off an USB drive rather than using optical disk that's possible of course. I have a USB HDD here I use for work that I customised to choose many different flavors of Windows OS install.