Learn Finnish: Tyttöystäväni kirjoitti kolme postikorttia.

Kirjoittaa is a Type 1 verb. We first take the stem, kirjoitta-. To make it past tense, we add the past tense marker -i. In verbs with three or more syllables, a final -a or -ä turns into -i (or simply drops out) when bumping into this past tense marker, giving us kirjoitti-. Because it's the "he/she" form, no personal ending is added, leaving us with exactly kirjoitti.

Why isn't there a separate word for "my" before tyttöystäväni?

The possessive suffix -ni attached to tyttöystävä (girlfriend) already means "my". You could write minun tyttöystäväni, but the pronoun minun is optional and often left out, especially in standard written Finnish.

If "three postcards" is the object of the sentence, why isn't it in the accusative (genitive) case?

Numbers boss other words around! Normally, a complete action's object might take a genitive-looking ending (like -n). However, any number greater than one requires the noun following it to be in the partitive singular. So, postikortti becomes postikorttia. The number kolme itself stays in its basic (nominative) form as the object.

I see tyttö and ystävä in the first word. Is this just a literal translation of "girlfriend"?