Partitive articles (Du / De la / Des / De l') are used when we talk about a not-specified quantity. They are a variation of the indefinite articles (un/une/des) and can be translated by the English word some.
J'achète trois œufs (I buy three eggs) → I know the exact amount of eggs.J'achète des œufs. (I buy some eggs / I buy eggs) → I ignore the exact quantity of eggs.The partitive takes the forms du / de la / des / de l'.
Le serveur apporte du pain sur la table. (The waiter brings (some) bread to the table.) → We don't know the exact quantity of bread, so we use the partitive : de + le = du pain.Il dispose des couverts pour chaque client. (He sets cutlery for each customer.) → We do not know the exact number of cutlery, so we use the partitive. : de + les = des couverts.Il ajoute de la sauce au milieu de la table (He adds (some) sauce to the middle of the table). → We don't know the exact quantity of sauce, so we use the partitive. : de la sauce.Puis il sert de l'eau aux clients. (Then he serves (some) water to the customers.) → We don't know the exact quantity of water, so we use the partitive. : de l'eau.In a negative sentence, you only need to put the preposition "de" between the negative adverb and the noun.
Il n'y a pas de pain sur la table. (There is no bread on the table)Nous n'avons plus d'eau dans nos verres. (We no longer have water in our glasses.)Je n'ai jamais de chance aux jeux de hasard. (I never have any luck at games of chance.)