"Beni" is accusative case and marks the (definite) direct object of a sentence.
"Bana" is dative case and marks the indirect object of a sentence.
English doesn't distinguish dative and accusative any more, merging them to a single "objective case" in pronouns.
However, sometimes English uses "to" in cases where other languages use dative -- for example, "He gave me a book" can be "He gave a book to me".
Note that sometimes, the only object of a verb is in the dative; this just has to be learned. For example, in Turkish, inanmak (to believe) and bakmak (to look at), among other verbs, take the dative for their object.
So you would say "Kadın bana inanıyor" for "The woman believes me".
You would also say "Kadın bana bir kitap veriyor" for "The woman is giving me a book".
One thing I don´t udnerstand... when teaching the continuous present in Turkish, why verbs are choosen that are exceptions in English. I want to learn Turkish and not English. When it is possible to say in Turkish the woman is hearing, but not in Engish, that it is just the wrong verb to teach beginners the Turkish continuous present. There are 1000s of verbs in Turkish (I guess) that are also accepted in English as continuous. So why choose the verb to hear then?