These are names for formulaic patterns in common-practice music. These patterns can then be used to compose/improvise phrases in the style of this music. These patterns are called schemata by psychologists and in the Gjerdingen book that Pat recommended, and partimenti within the historical Italian style. (It's more nuanced than that, but I'll let you research the distinction!)
The "Scalino 7.1," for instance, will have scale-degrees 7 and 1 in the bass. It's a part of a V6–I progression, and the upper voice is typically the interval of a third (tenth) above the bass on scale-degrees 2 and 3.
For these particular patterns, I recommend Georgio Sanguinetti's The Art of Partimento. Gjerdingen's book is great and discusses several schemata, but it seems that you're looking more for the partimento tradition.
As for the video, it seems that it's just a girl and her teacher trading off as they improvise phrases using these partimenti. But it goes to show you that training in this tradition really pays off in terms of improvisation!