How would you translate $\neg(P\to \neg Q )$ into English? I am having trouble in how to translate the $\neg(P\to \neg Q  )$ into English. I attempted first letting P and Q represent certain statements. For instance, let $P$ be the statement "I will gather the ingredients," and let $Q$ represent "I will cook a meal." Then the statement would represent:

"It is not the case that gathering ingredients is a sufficient
  condition for me to not cook a meal."(Correct me if I translated it
  wrong; this is why I'm asking this question)

One side note is that when you simplify the statement $\neg(P\to \neg Q  )$ it is equivalent to the statement $P \land Q$. Furthermore, when I translate using my example, it means "I will get ingredients, and I will cook a meal."However, that sounds like cheating since it does not feel like it represents the statement $\neg(P\to \neg Q  )$ . Anyhow, would that still be an acceptable besides my answer,"It is not the case that gathering ingredients is a sufficient condition for me to not cook a meal." Also, is my former answer a good representation for the statement $\neg(P\to \neg Q  )$? Do you know any alternative answers for the translation of $\neg(P\to \neg Q  )$?
 A: I think your "meal" example is getting in your way somewhat.
Let $P$ be the statement "It is raining today". Let $Q$ be the statement "I will go outside today". For a lot of people, $P \rightarrow \neg Q$ - "if it is raining, I will not go outside today".
I'm generally alright with rain, though, so this statement doesn't apply to me - its negation applies to me. $\neg (P \rightarrow \neg Q)$ - "it is not that case that, if it is raining, I will not go outside", alternatively "rain does not prevent me from going outside".
You wouldn't believe that this were true unless you had actually seen me go out in the rain, in the same way that the statement $\neg (P \rightarrow \neg Q)$ is equivalent to $P \wedge Q$ - "it is raining, and I am outside". If you knew that I was out in the rain, then you'd know that rain doesn't prevent me from going outside. To give credence to the idea that rain doesn't prevent me from going outside, I must have been out in the rain before, so yes, your former answer is a good representation for the statement $\neg (P \rightarrow \neg Q)$. "$P$ does not prevent $Q$" is another way to phrase it.
A: Yes, you're correct in your translations: both the direct translation, and the translation of the equivalent statement.
Your translation of the equivalent statement $P \land Q$ certainly sounds more direct, and it expresses, logically, exactly what your first translation conveys.
As an alternative to your first translation, you might instead want to say, directly from the given expression $\lnot (P \rightarrow \lnot Q)$, using the $P$ and $Q$ you designated: 

"It is not the case that, if I gather the ingredients, then I won't cook the meal." 

Maybe that sounds nicer to you than the sufficiency statement. But to get to the point: 
I think that your observation that the given proposition simplifies to the equivalent proposition: $$\lnot (P \rightarrow \lnot Q) \equiv P \land Q$$ gives you license to use the most direct of your translations: 

"I will gather the ingredients, and I will cook the meal."

A: What  is being missed here is the " implies relation"
 I would say Not the case such that P implies not happening of Q  ...ie

Incorrect to say that P prevents Q

Never has my not $\underbrace {cooking \quad meal}_Q$ been  decided by my ability to $ \underbrace {gather \quad  ingredients} _P$.
Never could  my success in gathering ingredients prevent me from cooking a meal.
Never does a Downpour prevent me from going outside.

So what we gather is: " A implies not B"  can be translated as 'A prevents B'
A: Given $P$, $Q$ is possible.
Or, $P$ doesn't rule out $Q$.
