Expand Help on the web
I was trying to find more information in the help function (https://help.vivaldi.com) to understand more how different functions worked. Did not find it. I suggest you expand the help function. I tried to find more information about: hibernate - how does it work? background tabs - what is it? tab contextual menu
The fact that there is "help" at all for a browser so early in its development, whose features are changing bascially daily through this development phase, is kind of remarkable. That said:
Hibernation is simply the caching of the tab as-is, and zero memory or processor cycles consumed to view it or update it until/unless you look at it again.
Background tab is a tab that is created and loads while you are not looking at it. If you right-click a link and pick "open in background tab" from the context menu, a new tab is created and the content from the link destination is loaded in it, but that tab does not get focus until you click on it. In other browsers, "open in new tab" always works this way, and none of them have an ability to open a new tab and focus it at the same time (the way the original "open in new tab" code was written over a decade ago). Vivaldi is the only browser that offers you the choice "open in new tab" (and focus that tab) or "open in background tab" (and keep browsing right where you are). Open in background tab can also be triggered with ctrl+click, middle-click, or a mouse gesture.