Learn Icelandic: Hann sýnir mér þykka bók.

The word bók is the direct object of the sentence (the thing being shown), so it is in the accusative case. Since bók is a feminine noun, the strong adjective þykkur (thick) takes the feminine accusative ending -a, becoming þykka.

Why is the word for me mér here instead of mig?

The verb sýna (to show) takes two objects. The person you are showing something to is the indirect object, which takes the dative case. Mér is the dative form of ég (I). The thing being shown is the direct object, which takes the accusative case (þykka bók).

The word bók didn't change form. How do we know it is in the accusative case?

Many feminine nouns, including bók, have the exact same spelling in both the nominative (dictionary) and accusative cases. Even though the noun itself doesn't change, the adjective attached to it must still take the accusative form (þykka).

How would the adjective þykka change if he showed me a masculine or neuter object instead?

If he showed you a masculine noun, like veggur (wall), the adjective would take the masculine accusative ending -an, making it þykkan vegg. For a neuter noun like teppi (blanket), it would take the neuter ending -t, making it þykkt teppi.