"your" = "dein" (masc.), "deine" (fem.), "dein" (neut.), "deine" (plural)
"your" (plural) = "euer" (masc.), "euere" (fem.), "euer" (neut.), "euere" (plural)
"your" (polite) = "Ihr" (masc.), "Ihre" (fem.), "Ihr" (neut.), "Ihre" (plural); ((notice they're all capitalized, b/c they're addressing someone in a formal manner))
So, it depends on who the 'your' is addressing when saying "Your children eat." (or "Your children are eating.") If addressing someone in an informal manner (a friend, close family member, etc.) = "Dein/Deine Kinder essen." If addressing several 'yours' (talking to both parents/guardians at the same time) = "Euer/Euere Kinder essen." If addressing someone in a formal manner (your parents' friends, authoritative figures, etc.) = "Ihr/Ihre Kinder essen."
This is as I understand it so far. Take a look back, close to the top of this thread, +AlexRttr has posted an excellent chart ('Possessive Adjectives') that has helped me a lot with my Deutsch 'your' questions. (I got all the info. I typed above, from that chart ☺.) Also, scroll down about 20 posts, and check out what +Elardus has to say about the use of your; some helpful info. there as well.
No, the second person singular forms - thou, thee, thy, thine, thyself - are archaic in English, almost never used nowadays, except for quoting old works of literature, sometimes in prayers or hymns addressing a deity (usually the Christian Father-god-aspect or Jesus-aspect), rare appearances in poetry, or for humorous effect (usually by people who do not know the correct archaic usage of the various forms). I understand the Quakers continued to use some of the forms after they fell out of use in the general population; I do not know whether this is still the case in some situations, although I would guess it is not; they use(d) "thee" for both subjects and objects, I believe, and modern third person singular verb endings instead of the traditional second person verb endings - "Thee is reading a book," instead of "Thou art reading a book."
That said, in the old days when "thine" WAS generally used, it still would not have been used in a sentence like this; it would not be used before a noun starting with a consonant like "children", but before nouns beginning with a vowel: "thine uncle" but "thy children". ("thine" was also used in a few other places as a possessive pronoun standing alone, not followed by a noun, as in "That book is thine" or "That is no concern of thine."
- du = singular "you" and from that you get "dein" and "deine" (your) when used with different nouns, as I'll explain below.
- er = "he" and from that you get "sein" and "seine" (his).
- sie = "she" from which you get "ihr" and "ihre" (her).
- wir = "we" (or "us") from which you get "unser" and "unsere" (our).
- ihr = plural "you" from which you get "euer" and "eure" (your, plural).
There's also "Sie" with a capital "S" which is the formal "You" and applies to both singular you and plural you and it works the same as "sie" (her), that is - you get "Ihr" and "Ihre" from it.
Now, when do you use one of the other? It depends on the gender of the following noun. If it's masculine or neutral, you use the first one, that is: dein, sein, ihr, unser and euer. Example: "Das ist dein Kind" (This is your child) because it's "das Kind" (neutral). "Das ist ihr Stift" (This is her pen) because it's "der Stift" (masculine).
If the noun is feminine or if it's a plural, then you'll use deine, seine, ihre, unsere and eure. Example: "Das ist seine Mutter" (This is his mother) because it's "die Mutter" (feminine). "Das sind unsere Kinder" (These are our children) because it's a plural "die Kinder".
A simpler way to think of it could be - if the noun is a "der" or "das", it's the first option, but if it's "die", it's the second.