English translation:
never
by no means
impossible
Structure:
決して + Noun・ ではない
決して + Verb[ ない ]
決して + いAdj[ くない ]
決して + なAdj・ ではない
Explanation:
[There are two ways to read 決して: けっして and けして. The first one is the original reading, and is used the most. けして can be used in more casual situations, but some might consider it a mistake when used in writing.]
OK ! So I have a question… when this grammar point comes up, how do I know if it’s **決して they asking me to use, 全然(ぜんぜん) or 全く(まったく), since both of them (all three of them, actually) have ない at the end of the sentence? And they all seem to mean approximately the same thing…
I tend to use ぜんぜん by default… would be a mistake to say one instead of the other?
thanks!
とても
Hey 
Sorry for the late answer 
They are very similar since they all express total negation but:
全然 and 決して are not used in positive sentences like (though the younger generation uses ぜんぜん in positive sometimes):
それは全く合理的な決意でした
それはとても合理的な決意でした。
That was a very rational decision.
Next thing ,
決して、とても and 全然 cannot modify a noun with の like:
かれの話はまったくのうそだよ！
(His story is complete bullshit)
決して is very often used when forbidding something.
とても is mostly used when denying the possibility of something, so it is often used with potential form.
アルコールを飲んだら決して車を運転してはいけません。
You must never drive after drinking. (IMO, it has a slight nuance of tenacity)
オレにはとても歌えない。
It is completely impossible for me to sing. (I really suck at singing.)
宝くじを当てるなんとてもてありえないよ。
Something like winning a lottery, absolutely impossible.
In terms of formality (from least to most):
とても ない ＜ ぜんぜん ない ＜ 全く ない ＜ 決して ない
And 全く ない and 決して ない sound too formal for friendly conversation.
So using ぜんぜん as default is OK.
Thanks so much, this is so useful! I had no idea 全く ない was also too formal to use with friends. I think it’s more clear now, I’ll try it out!
Though まったくby itself is used pretty common though.
For the example “I will never give up”, wouldn’t “絶対諦めない” also work?
Also, why don’t we have a grammar point for ぜったい? 
Can this grammar point work with negative desire form? I.E; 決して諦めたくない。I don’t see why not but I’m just confirming here.
Straight from the writeup:
As an adverb, it carries the meaning of ‘decidedly’, or ‘assuredly’, and is used in sentences that contain ない, in order to convey ‘never (A)’, ‘under no circumstances (A)’ or ‘by no means (A)’.
So yeah! It can be used with ～たくない. The key here is for the verb to be in its ない form.