"No, that is not right."
違う (ちがう) is a verb meaning: (1) to differ (from), to vary; (2) to not be in the usual condition; (3) to not match the correct (answer); (4) to be different from promised. Source: https://jisho.org/search/%E3%81%A1%E3%81%8C%E3%81%86
The ~います ending is a form of the root verb (one drops the ~う when conjugating a verb).
ちがう is dictionary form and ちがい is another word entirely, if it's even a word. The problem with duo is it splits the conjugation into seperate tiles, making the different parts of words look like seperate things. All る verbs end with る and all う verbs end in an う syllable (う, く, む, つ, す, etc.) Standard conjugation is remove る for る verbs, and change from う syllables to い syllables for う verbs, then add ます for present tense positive or ません for present tense negative. ねる(sleep) becomes ねます(to sleep) or ねません(to not sleep). さがう(be different) becomes ちがいませ(to be different) or ちがいません(to not be different). Obviously there are other forms of conjugation, but I'm still learning and those forms haven't been covered yet at this point in the lessons.
1 is for saying a fact is wrong e.g. これはリンゴですか - 違います、ナシです (It is about a fact is right or wrong, so そうじゃないです/そんなことないです cannot be used)
2 is for saying a method/happening is wrong (referred by そう - that way) "It is not that way" e.g. 公園に行きますか - そうじゃないです（違います is also ok because it is also a fact）、銀行に行きます (It is something future, so そんなことないです cannot be used)
3 is saying if there is such a happening "そんなこと" or fact. It is not right or wrong, it is about a fact exists or not. きれいですね。 - そんなことないです "What you said doesn't exist" (much softer than 違います which says it is wrong)
It is the non-past polite conjugation of the verb 違う meaning "to differ"
違います - "it is different" (that thing is different than the correct thing, it is not right)
違いです would use the noun form 違い meaning "difference, discrepancy, error" with the copula です(is/am/are) and would mean "It is an error/it is a difference", which has a slightly different use.
ちがいます is the polite form of the verb 違う (ちが.う), which means "to differ (from)". Negating it would imply that whatever's being talked about "does not differ from" whatever the assumed or described state is, but it seems to be somewhat rare to use the word that way. If you instead wanted to reply in the affirmative here, you would likely just use a different word, or simply はい.
I'm not a native speaker (so take what I say with a grain of salt), but if you search for 違いません you'll have a hard time finding example sentences, which is usually a sign that whatever phrase or construction you're searching, if not grammatically incorrect, is at least something that just isn't done for one reason or another.
For example, if you search weblio (https://ejje.weblio.jp/content/%E9%81%95%E3%81%84%E3%81%BE%E3%81%9B%E3%82%93), you'll actually be presented with examples of 間違いありません.
間違い (まちがい) here is a noun and means mistake or error, and ありません is the polite negative form of the verb ある (to be; to exist for inanimate objects). It translates to "There is no mistake." or "I'm sure." depending on what the context is.
Sorry for going off on a bit of a tangent, but it's helpful to try looking up words or phrases you're unfamiliar with to find examples in context. Duolingo is particularly bad about providing context in general, so things end up being more confusing than necessary fairly often.
Anyway, hopefully that answered your question.