"He knows this girl."
Translation:Он знает эту девочку.
I think your confusion probably comes from masculine nouns where the accusative of inanimate nouns matches the nominative, and the accusative of animate nouns (people, animals) matches the genitive.
Он знает этот дом
Он знает этого мальчика
Both are in the accusative, but they don't look like it.
(Edited to fix этот to be correct - it also matches the nominative/genitive in the same way)
How is animate/inanimate decided? In, say, a fairy tale with a living house (wasn't there something about "Baba Yaga's walking house on cjicken feet"?), would you refer to that particular house as "этого дома"?
How about Stephen King's Rose Red, a manor that does not move or talk itself, but that seems to "have" a mean-spirited personality, trapping people inside?