Learn Finnish: Minä tunnen heidän perheensä.

In the third person (his/her/their), standard Finnish requires both the genitive pronoun (heidän for their) and the possessive suffix at the end of the noun (-nsä). You cannot drop heidän here, even though you are allowed to drop the pronoun for my or your.

Wait, shouldn't the object of tunnen end in -n? Why is it just perhe + -nsä?

You are right that the verb tuntea (to know someone) normally takes a total object ending in -n (like perheen). However, there is a special rule: when you add a possessive suffix to an object, the -n ending disappears! So the total object perheen plus -nsä simply becomes perheensä.

Why do we use tunnen instead of tiedän for I know?

Finnish has two common verbs for to know. Tietää is used for knowing facts or information. Tuntea is used when you know a person, a family, or are familiar with a place. Since we are talking about knowing people, tunnen is the correct choice.

Why does the suffix use an ä (-nsä) instead of an a (-nsa)?