What is the relationship between Natsu and E.N.D? From what I know, E.N.D is the most powerful demon created by Zeref.
Zeref tells Natsu to choose between life and death. He's practically telling him that he'll have to decide if he wants to live or if he's going to kill himself for the sake of the other’s. He'll want to die instead of hurting his friends.

Zeref said that Igneel wanted to kill E.N.D but couldn't.

So, What is the relationship between Natsu and E.N.D?
In chapter 436 it is revealed that, ...


   Natsu (Dragneel) is the younger brother of Zeref (Dragneel). Zeref's younger brother died a few years after birth. Unable to let go, Zeref researched everything he could find about bringing back the death (R-system), travelling through time (Eclipse Gate) and creating magical beings (Etherious). In the end (pun not intended), he combined all of his knowledge and created an Etherious version of his younger brother, using the original body of that younger brother, named E(therious) N(atsu) D(ragneel).

 

 On a side note. This does not yet explain why Zeref was holding the book of E.N.D. in chapter 413. By holding the book, it would give the impression that E.N.D. is not yet summoned, which imho contradicts the above.