The first time I met her, I noticed how good a listener she was:

Onunla ilk tanýþtýðým zaman ne kadar iyi bir dinleyici  olduðunu farkettim.

Soru:

Bu cümledeki "HOW" sözcüðü, Türkçe dil bilgisinde nedir?Bir "adverb" mi, bir "preposition" mu, bir "conjunction" mudur?

Hiçbiri. Soru sýfatý. Ýngilizcede "question words" deniyor.

The first time I met her, I noticed how good a listener she was:

Onunla ilk tanýþtýðým zaman ne kadar iyi bir dinleyici  olduðunu farkettim.

I will try to explain what ´how´ is, in this sentence.

Normally ´how´ is a question word.

But, in this sentence it is used differently.

Its meaning is like:

The first time I met her, I noticed what sort of listener she was:

Onunla ilk tanýþtýðým zaman ne çeþit bir dinleyici  olduðunu farkettim.

She was a very good listener (onu bir çok iyi dinleyici oldu ?)

´How good a listener´ means ´she was very good´

Mustafa Sandal´s song ´Araba´ had a line

güzelmi güzel which translates roughly to

´how beautiful is the beauty´ or

´it is very beautiful´ 

Your sentence is not asking how good she is, but implying that she is very good at listening.

So here, I think ´how good a´ is like an adjectival phrase,

(sýfatlýk ifade?), that describes the ´listener´

I hope this is not too confusing, and helps you understand your sentence better. 

(Umarým beni kafasýný karýþtýrdýn)  ? Is this correct?

...

Quoting Henry

I hope this is not too confusing, and helps you understand your sentence better. 

(Umarým beni kafasýný karýþtýrdýn)  ? Is this correct?

 Umarým kafaný karýþtýrmadým.

Literally: I hope I haven´t confused your head.

It is karýþtýr-ma-dým because you want to say the negative (to not confuse) and the person doing the confusion is "me".

It is kafaný because you want to say your head, not his head.

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Many thanks for your thorough explanation Marion. 

"The first time I met her, I noticed how good a listener she was:

Onunla ilk tanýþtýðým zaman ne kadar iyi bir dinleyici  olduðunu farkettim.

Soru:

Bu cümledeki "HOW" sözcüðü, Türkçe dil bilgisinde nedir?Bir "adverb" mi, bir "preposition" mu, bir "conjunction" mudur?"

"good listener": iyi dinleyici

good/iyi: sýfat (adjective)

listener/dinleyici: isim (noun)

How (good): Here ´how´ means ´very, much´. It reveals the degree of ´good´ and therefore it is an adverb in Turkish grammar.