Learn Romanian: Pisica doarme pe perna mea. - The cat is sleeping on my pillow.

In Romanian, when you use a possessive adjective like mea for my, the noun before it must have the definite article attached. Pernă means a pillow, while perna means the pillow. You are literally saying the pillow my.

Does the preposition pe change the case of the noun perna?

The preposition pe requires the accusative case. Fortunately, for nouns in Romanian, the accusative case looks exactly the same as the nominative form used for subjects. So perna remains unchanged here.

Why does the verb root in a dormi change to doarme?

In the third person for he, she, or it, the root vowel o changes to the diphthong oa because the stress falls on that syllable. This shift from o to oa is a very common sound change in Romanian irregular verbs.