"The child is fine."
Translation:У ребёнка всё хорошо.
This is confusing! I guess I don't understand why one would describe the child as having all right/fine, instead of being all right/fine. Is it saying that the child has/possesses a certain state of being, in this case, the state of being fine?
Yes. This is equivalent to English "Everything is fine with the child", just without "is" - which is almost never used in this kind of statements in Russian.
No, it's grammatically incorrect. Хорошо is an adverb and hence cannot modify nouns. Hence grammatically it should be either "Ребёнок хороший" (the child is good) or "У ребёнка всё хорошо" (everything is well with the child). In both cases the Russian equivalent of "is" is dropped from the sentence, but is implied grammatically anyway. So in the first case "хороший" is an adjective used to describe the noun (ребёнок), while in the second case "хорошо" is an adverb (well) used to describe the verb -- the missing "is".
The meanings of these two sentences are quite different in both Russian and English though.
Note that in general Russian is much stricter than English (esp. American English) in distinguishing adverbs and adjectives - the Russian analogues of "good" and "well" are always distinct words with distinct grammatical functions.
P.S. You can also say "Ребёнку хорошо", meaning "the child is feeling well". Literately this sentence should translate into something like "All is well to the child", with both the subject (all) and the verb (is) not explicitly present in the Russian sentence in this case.
The mouseover information is not very helpful here, as it basically says it should be what @mpafp said.
Good explanation. I got this wrong and got the suggested answer "Ребёнок в порядке" (the child is in good health) as well as the answer above. Is this a common phrase?
"X в порядке" literally means "X is in order". It does not mean "in good health" but it does mean that you should not worry about him/her/it. It's definitely not an uncommon construction, and "всё в порядке"="everything is in order/fine/OK" is actually very common.