die
The German word 'der' is the definite article for masculine nouns in the nominative case. It is equivalent to the English word 'the'. It is used to indicate a specific instance of a noun, as opposed to an indefinite article like 'ein' (a).
The table is made of wood.
In this sentence, 'der' is used as the definite article for the masculine noun 'Tisch' (table). In German, each noun has a gender (masculine, feminine or neuter), and the definite article changes according to the gender of the noun. Here, 'aus' is a preposition that denotes the material of the object (table), followed by 'Holz' which means wood.
The dog is playing with the ball.
This sentence uses 'der' as the definite article for the masculine noun 'Hund' (dog). The verb 'spielt' means 'plays' and 'mit' is a preposition that means 'with'. 'Dem' is the form of the definite article 'the' used before a dative masculine or neuter noun. 'Ball' is a masculine noun in the dative case here, meaning ball.