Omg, did I spoil you?! I’m so sorry :/// (that’s why people shouldn’t be reading my tags) 

Okay, um, the thing is, the movies are leaving a lot of things to the side (like Guts’ backstory, a lot of character developments and plot-related moments) and the child thing should have happened at the end of the third movie but they made Guts leave and ended it there.

I’ll try to really give you a short summary to avoid spoiling you too much because this is also relating to vol 1-2-3 of the manga which are post-eclipse too (and the anime this summer will probably begin by them).

So basically, when Guts and Casca had this romantic moment before they went to rescue Griffith, it turns out they actually conceived a child but, with what Griffith (or Femto) did to Casca during the eclipse, the child got cursed and turned into a demon.

Casca (still mentally affected by everything she went through) “gave birth” to it a few days after the eclipse (or you could see it as a miscarriage, except the little thing is alive) in front of Guts and the Skull Knight.

And that’s really all I can tell you without spoiling more since afterwards Guts leaves on his quest for revenge (end of third movie).
The Skull Knight confirmed the child had been cursed by Griffith and not conceived when he raped her, meaning that it’s Guts’ child for sure, and Guts tried to kill it but Casca wouldn’t let him (until it disappeared at dawn like the other dark creatures which led her to cry a lot about this).

After that…. well, you’ll see in the anime of this summer 😉 the child is an important element because, quoting the Skull Knight…

and, as it is often the case, he’s right. 🙂

Sorry again for spoiling you Anon I hope my answer will be enough, you will see more about it if you watch the anime this summer (if it respects the manga’s plot of course).

Have a nice day 🙂

(PS: in case you don’t mind spoilers, if you want a reason to read 38+ manga volumes and if you’re interested, under read more is one popular (?) theory within the fandom about Guts and Casca’s child)

And it is that this little demon above (cursed by Griffith’s seed when he raped Casca) turned into…

a little boy who appears only when it’s the full moon because… Well, because something happened involving Guts, Casca and the God Hand after Guts became the Black Swordsman (most likely the arc will get covered by the 2016 anime starting this summer).

Seriously Anon, all I can say is read the manga: amazing art + amazing plot + lots of action & character developments and it’s a million times better than the movies (I started with the movies too and I wasn’t sure I wanted to read 38 volumes but it’s 100% one of the best decisions I’ve taken! ^^)

Sorry for all the rambling! I hope it answered your question at least. 🙂