While the partitive exists in English, it goes mostly unused. Like so many other things in English, we understand it by implication. But as with so many other languages, implication alone isn't good enough for Italian grammar.
What is a partitive?
The partitive is a special kind of pronoun which functions as a back-reference. It refers back to a direct object specified in a previous sentence - or possibly later in the current sentence.
Observe the following examples:
Do you want some coffee?
No, thanks. I don't want any (of it).
Would you like some pie?
Yes, I would love a piece (of it).
In both of the examples, the partitive "of it" is generally omitted in English. We understand the partitive by implication. But in Italian, implication is not enough. Grammatically, a noun or pronoun must be specified. And that's where ne comes in.
I've had enough "of it"!
The Italian word ne is a partitive pronoun. That is, it is a noun which refers to another noun, or a portion thereof.
Quante birre hai bevuto ieri sera?
Ne ho bevute quattro.
Vorrebbe del caffè?
Grazie, no. Non ne bevo.
In the first example, when asked how many beers were drunk the night before, the response is "ne ho bevute quattro", or "I had four (of them)." Also an important thing to note here is the way that using ne requires the participle to agree in gender with the object noun being referenced. Hence bevute rather than bevuto.
In the second example, the response to an offer of coffee is "non ne bevo", meaning I don't drink it. The "of it" isn't immediately clear, but when you consider the way del is used in Italian to mean some, it becomes more clear. Del literally means "of it", even if it's more comfortable in this case to translate it differently, so here it helps to think of ne meaning "any of it".
Cosa ne pensate?
I think it should be starting to become clear that ne is, basically, a back-reference to di + a noun. And since there are a few other ways that del can be used, it's no surprised that there are also a few other ways that ne can be used.
Ho conosciuto Maria e ne ho visitato la casa.
Andiamo al cinema, che ne dice?
È un uomo eccezionale, tutti ne parlano con ammirazione.
In the first example, ne means di Maria, and it completes the possessive la casa di Maria. However it's split because ne belongs grammatically at the beginning of the phrase.
In the second example, we can more readily envision the missing English partitive: "... what do you say (about that)?" Even though it translates better as "about that", it is once again a case where the Italian words being replaced are di + a noun. Cosa dice di questo?
And in the final example, we once again translate the missing partitive more comfortably as "about him", but it's a little easier to make the connection when you think of it as "everyone speaks of him with admiration."
So there it is: the partitive, a fascinating grammatical feature. (And one that makes a lot more sense after a short explanation than it would if you spent weeks trying to figure it out from conversation!)