The partitive article in French is the English equivalent of ’some’ and ‘any’.  So, it is very often used and thus, very important to learn.  First, let’s talk about the form:

Before a masculine noun - du
Before a feminine noun - de la
Before a masculine or feminine plural noun - des
Before a masculine or feminine singular noun beginning with a vowel or silent h - de l’

And here is how you use them:
Before a concrete or abstract noun to indicate an uncertain quantity or a part of a whole that can not be counted.  In other words, with non-count nouns.  For example:
Je mange toujours du pain avec les repas. (I always eat bread with meals.)
Je vais prendre de la monnaie.  (I’m going to take some money.)
J’ai mangé des carottes.  (I ate some carrots.)
Je bois toujours de l’eau avec les repas.  (I always drink water with meals.) 

Remember, the partitive article is often used with the verb faire (to do): faire du vélo, faire du sport, faire de la guitare, faire des études, faire du théâtre, etc.

Be careful as you may forget to use the article as it is often omitted in English! 

Also, use only de after adverbs of quantity like beaucoup and moins.  For example,
Je bois beaucoup d’eau avec les repas.  (I drink a lot of water with meals.)
Je mange moins de pain les weekends.  (I eat less bread on the weekends.)  

And, use only de in negative sentences.  For example,
Je ne bois pas d’eau.  (I don’t drink water.)
Je ne mange pas de pain avec les repas.  (I don’t eat bread with meals.)
Je ne vais pas prendre de monnaie.  (I’m not going to take any money.)