"The hospital is over there."
The "over there" that むこう refers to is more like "across from". As in, "over there, across the street". あちら is less specific but refers to a direction that is neither towards the speaker (こちら) nor the listener (そちら). I feel this might be a poor explanation so if someone else can chime in that might be helpful. :-/
Translations are not exact. あちら may translate to 'there' (an adverb) in English, but in Japanese it functions as a pronoun. Likewise, むこう functions as a noun. A similar thing can be seen with words such as そば (beside) and 上 (above) which are nouns, but translate as prepositions into English. However, you can turn あちら (and そば, and 上) into an adverb by adding に, to make あちらに.
あります was incorrect. Duo expected です What would the difference in meaning be with one over the other? Would あります be more like "Over there is a hospital"? Sorry, beginner so some of this less-explained stuff (like when to use aru/iru vs desu). Is it desu here because "over there" is a noun?
I'm curious now as well. I tried using あります and was considered wrong but I'm not 100% sure if it is simply because Duo hasn't allowed it as an answer or if it is actually incorrect to say.
です = To be / Is (A = B)
ある/いる = To exist (A exists)
The [hospital] is [over there] = です
The [hospital] exists [over there] = あります
It shouldn't really matter which you use when talking about location, right?