1. A partitive article
Du, de la, de l’, des

2. Du, de la, de l’ = unspecified singular quantity
It’s the notion of “some” in English, but we don’t always use the word “some”.
When you are talking about a portion of one item (food, like "some bread"), or something that cannot be quantified (quality, like "some patience"), use what the French call " a partitive article":

3. Ex: Je voudrais de l’eau, s’il vous plait
Ex: Je voudrais de l’eau, s’il vous plait. (some water, maybe a glass, or maybe a bottle…)
Ex: Le professeur a de la patience. (patience ; you are not saying how much patience the teacher has, just that he/she has some)
Ex: Voici du gâteau. (some of it, not the whole cake)

4. du is used for masculine gender
Je veux du café De+ le =du
I want some coffee.
Also we say: Je prends de l’eau. If the noun starts with a vowel
I am taking (some) water.

5. Vous prennez de la limonade.
De la feminine gender is not contracted de + l’ if the noun starts with a vowel
Tu prends de la soupe.
Vous prennez de la limonade.

6. Ils prennent des croissants. They are taking (some) croissants.
If we want express plural we will use indefinite article plural – des Des = more than one, but unspecified plural quantity
Ils prennent des croissants.
They are taking (some) croissants.

7. Ex: J’ai des Euros. (more than one, but I am not telling exactly how many)
Ex: Je vais acheter des pommes. (I’m going to buy apples. In English, we’d probably won’t use any words before "apples". Maybe some, but not necessarily. In French, you need to use “des”)
Ex: Elle a des amis formidables (she has (some) great friends)

8. Be careful though with translation.
In English, the word “some” is used for unspecified quantity (I would like some milk) but also as a derogative adjective (he went home with some girl).
In French, you would never say “il est rentré chez lui avec de la fille”…He didn’t go home with an unspecified quantity of a girl. So be careful, translation doesn’t always work!

9. Specific quantities Un, une = one and the numbers.
This one is quite easy. When you are talking about a whole item, use:
un (+ masculine word) to say one. Ex: J'ai un fils (I have one son).
une (+ feminine word) to say one. Ex: j'ai une fille ( I have one daughter).
a cardinal number, like deux, or :-) Ex: j'ai deux filles (I have two daughters).
Note that "un and une" are also "indefinite articles" in French, meaning "a/ an" in English. 

10. More specific quantities = expressions of quantity are followed by de or d' !
Expressions of quantity are followed by "de" (or "d”), never "du, de la, de l', or des".

11. In English, you say "I would like a little bit OF cake", not "a little bit SOME cake" don't you? Well, it's exactly the same thing in French. So, in French, after an expression of quantity, we use “de” or “d'” (+ word starting with a vowel).

12. examples
Ex: Un verre de vin (a glass OF wine, NOT DU, you do not say “a glass some wine”)
Ex: Une bouteille de champagne (a bottle of champagne)
Ex: Une carafe d’eau (a pitcher of water – de becomes d’ + vowel)
Ex: Un litre de jus de pomme (a liter of apple juice)
Ex: Une assiette de charcuterie (a plate of cold cuts)
Ex: Un kilo de pommes de terre (a kilo of potatoes)
Ex: Une botte de carottes (a bunch of carrots)
Ex: Une barquette de fraises (a box of strawberries)
Ex: Une part de tarte (a slice of pie).

13. And do not forget all the adverbs of quantity, that also specify quantities :
Ex: Un peu de fromage (a bit of cheese) Ex: Beaucoup de lait (a lot of milk). Ex: Quelques morceaux de lards (a few pieces of bacon).

14. LA NÉGATION Always use ne ……pas de
Ex. Est-ce qu’il y a du lait? Is there (any) milk?
Non, il n’y a pas de lait. No, there isn’t (any) milk.
Est-ce que tu veux de la soupe? Do you want (some) soup?
Non, je ne veux pas de soupe. No, I do not want (any) soup.