"The horse eats about ten apples."
Translation:Das Pferd frisst etwa zehn Äpfel.
Yes, "almost" and "nearly" are pretty much synonymous; you could translate them both with fast.
etwa, on the other hand, means "approximately" or "about".
If you used fast zehn Äpfel, you would indicate that the horse definitely ate fewer than ten apples -- maybe it ate nine whole apples and part of a tenth one, so it was "almost ten apples" or "nearly ten apples".
While if you said ungefähr zehn Äpfel or etwa zehn Äpfel (about ten apples), it means that the number was either ten or close to it -- could be nine, could be eleven. Perhaps you weren't counting exactly.
If the horse in fact ate eleven apples, then "about ten apples" (ungefähr zehn Äpfel) would be appropriate but "almost ten apples" (fast zehn Äpfel) would not.
Äpfel is the form used in the nominative, genitive, and accusative cases, while Äpfeln is used in the dative case, in the plural.
When we just talk about “the plural form”, we mean nominative plural.
But almost all nouns take an extra -n ending in the dative case in the plural.
the word pferd
There is no word pferd in German. "Horse" in German is Pferd -- note the capital P.
Why do we use das
Because the word Pferd is grammatically neuter. And it's the subject of the verb in this sentence, so it's in the nominative case.
So we need the neuter nominative article das in front of it here.