"A man likes his daughters."
Translation:Ein Mann mag seine Töchter.
The daughters are a direct object of this sentence, and thus take the accusative case. The plural forms of "sein" are seine (Nom), seine (Akk), seinen (Dat), seiner (Gen). If they were the indirect object of the sentence - such as if he handed them a book, for example - it would be "A Mann gibt deinen Töchter ein Buch" (or "A Mann gibt ein Buch deinen Töchter" depending on what you wanted to emphasize). To get to "seiner Töchter", we need the daughters to be in the Genitive case - which, frankly, you probably don't need to worry about, as it's not used so much in spoken German these days.
I think I've been over weighting the value of this genitive case. I still don't fully understand when something is genitive (a part from the prepositions that trigger the case) to me this sounds as though it'd be genitive because they're "his" daughters. How does this differ from a situation where we do use the genitive case?
An example with genitive would be "Der Mann isst den Apfel seiner Töchter" - the man eats his daughters' apple. Apple is a direct object (what is being verbed), so it is accusative; the daughters own the apple, so they are in genitive.
You'd use "gefallen" to indicate that something is pleasing, which is different than "to like".
Thanks for the reply, daver3k. I don't think that's right, though: "is pleasing" gets the grammatical structure right, but "Das gefällt mir" is arguably the most common way of saying "I like that". In terms of meaning, then, gefallen has more to do with liking than pleasing.