"Eleni is drinking water."
Translation:Η Ελένη πίνει νερό.
so we always say "the" even if it's a name that's the subject? Instead of "Helen is drinking water", we write "The Helen drinks water" in Greek?
Literally, yes, it's "The Helen drinks water".
But you'd translate it as "Helen" and not "the Helen" because the English rules for articles are different.
Personal names always take the definite article: Η Ελένη βλέπει τον Γιώργο "Helen sees George", literally "the Eleni sees the Yiorgos".
(Exception: No article when you call someone by name. Ελένη, έλα εδώ! "Helen, come here!". And none if you are referring to the name itself rather than the person with that name: Το όνομά μου είναι Κώστας "My name is Costas", since there you're referring to the word rather than the person.)
νερό is the modern Greek word.
ύδωρ is the ancient Greek word which is sometimes still used, but much less often then νερό.
If you're in the Basics skill, notice that at the beginning (homepage) there is an icon of a bugle (I think) next to the icon of a key. The second (bugle) icon is for "tips and notes", and there is an introduction to the definite and indefinite article there. There may be similar information when you go to "discussion" where they have what they call "sticky notes" that are always there for your information.