These are all variations of the same thing, but they're being applied to nouns of different genders. They're basically contractions, so instead of saying "de la", they're run together to make "della". "de le" becomes "delle" "de i" becomes "dei", "de il" becomes "del" "de gli" becomes "degli".
So, della donna = of the woman. delle donne = of the women, del uomo = of the man, degli uomini = of the men, dei ragazzi = of the boys
Now, as hkysonjr mentioned, this same form can also be used to indicate "some". So, "delle mele" can mean "some apples", and it's the context in the sentence that will make it clear.
Nope, you're not used to listen carefully, i hear it perfectly, italians even speak faster (like ten times faster) you can't say they are wrong. Too much English for you for now. But I think this is more cultural than linguistic issue. By the way I'm not Italian nor English native. If you really want to complain about pronunciation try Danish. I will give you the reason by then.