тебя is the genitive version of ты, as you can see here http://www.russianlessons.net/grammar/pronouns.php
To the native English speakers:
Is it wrong to say: "The woman has already an apple"?
I don't know why, in this course I am always corrected when I make a sentence like that. Apparently Duolingo developers for the Russian course think that it's completely wrong to put ALREADY after the form of the verb TO HAVE. Are they right? Am I wrong?
Yes, it's wrong to say "The woman has already an apple" If you wanted to use this order you could say: "The woman has already (past tense verb) an apple" e.g. got/received/bought/taken/eaten/been given ...and this would change the meaning of "has" from possession (having something) to indicating a past action (having done something)
@Jana415365 - They do sound the same, as the stress is on the first syllable so the "o"'s sound like "ah"' s.
Check out declensions for words on Wiktionary, they put the stress markers on the words and it can help a lot: https://en.m.wiktionary.org/wiki/%D1%8F%D0%B1%D0%BB%D0%BE%D0%BA%D0%BE
Your first sentence is an example of negation using нет. Нет basically acts like не есть, and in Russian grammar rules this type of negation requires the use of genitive case.
In your second sentence, there is no negation. Яблоко can be considered the regular, nominative subject of the sentence (if you were doing a literal translation to English, like, "An apple is by the lady").
In nominative case, женщины is plural (The women, or Women). Here, we have the preposition У (meaning at/by/near, or to show possession) which requires genitive case. Female gender endings change from -a/-я to -и/-ы in genitive singular, so Женщина becomes Женщины. If it were many women having an apple, then it would be У женщин (the ending drops off) есть яблоко.