If an upgrade to Windows 10 is carried out on a system that was previously running a qualifying operating system, (usually Windows 7 or 8), the authentication process is carried out automatically by carrying across the authentication of the original operating system. There is no need for the user/upgrader to enter any codes.
If for some reason the original operating system does not qualify for an upgrade, or is not recognised as being genuine, then a nagging prompt will appear indicating that the Windows 10 installation is unlicensed. The screen background is locked and some other personalisation options are greyed out. That is the sum total of the difference between a licensed and unlicensed copy of Windows 10. You can of course change the screen background by just selecting a photo, right clicking and selecting the option to display that as the desktop background.
All updates and other functions should work fine.
Any PC with a genuine copy of Windows, built in the last decade or more ago will have a hologram sticker somewhere in or on the machine with the authentication code for that copy of Windows. A duplicate of the information should be contained in the documents supplied with the computer. If there isn't a hologram sticker or you haven't got the documents with the information, then the computer supplier didn't supply you with a valid copy of Windows in the first place.
There is a grey area where OEM copies of Windows software were used on multiple machines and not strictly in accordance with Microsoft's licensing requirements. Given that Microsoft currently demand between $120-$200 USD for a licensed copy of Windows depending on whether you want the Home or Professional version, you can see why a lot of constructors sailed very close to the wind from a legal point of view.
Basically Microsoft have made it uneconomic for pirated copies of Windows to be sold, because the genuine article works fine even if unauthorised. That is a good thing from the point of view of trying to ensure that computers users don't succumb to viruses and malware because they are running hacked versions of Windows. It makes good business sense to try and keep a degree of confidence in the security and stability of your operating system. Microsoft make enough money from the business market that they don't need to fleece home owners too, but there is no harm in trying, right?
So provided you can ignore the nag prompt and you are prepared to work around the personalisation limitations, an unauthenticated version of Windows 10 will work completely normally.
Although the licence number hologram stickers are usually stuck on the outside of computers, on the back panel of desktops and underneath the base of laptops, they may also be found stuck inside removeable panels such as inside the battery compartment or inside the upgrade panels for disk drives and memory, especially with laptops. If your computer was initially supplied with a genuine copy of Windows, the licence number will be on it or in it somewhere, but it may take some looking! :-)