"The woman is eating an apple."
Translation:Женщина ест яблоко.
Since Russian language doesn't have articles, their meaning has to be expressed by other words. Эта (Eta, this) is used to point out that we are talking about a specific woman. Leaving out the word would change the meaning of the sentence to "A woman eats an apple", at least when we have it out of context.
The pronunciation is the same, you understand it based on the context. Эта is feminine, it means "this" when talking about someone or something in particular; это can be the same, but with neuter nouns, or "this is", and it's used with all genders; for example: this is a car, this is my friend, this is a school... -> это; this car, this friend, this school -> эта (f), этот (m), это (n).
The accusative case of яблоко is яблоко. I guess it's an exception. Hope this helps: https://en.wiktionary.org/wiki/%D1%8F%D0%B1%D0%BB%D0%BE%D0%BA%D0%BE