이다 , 입니다 , 습니다/ㅂ니다 , 이야 and 이에요/예요. 和 나는 미국인 이야 and 나는 미국인 이에요 are they the same thing? 的差別在哪裡？如果不好說明，請提供一些例句。

A certified korean tutor explained this to me very nicely. I had asked him about difference between 이다 and 있다 and i think part of his answer will answer your question as well. I had also asked the tutor to confirm if '무슨 일이요?' and "무슨 일이에요?" both mean the same thing - as i keep hearing '무슨 일이요?' a lot in korean historical dramas and i think his below answer might help clear your confusion about 이다 -> 이에요/예요 -> 입니다

**Tutor's answer - 이다 -> 이에요/예요 -> 입니다. (you use '이다' when you say 'something is something'.)

있다 -> 있어요 -> 있습니다. (exist, there is, have)

이다 and 있다 are the most important predicates in Korean, they are completely different. VERY IMPORTANT.
이 is the stem which contains the core meaning from the base form '이다'.

** Tutor's answer - 무슨 일이야? = 무슨 일이요? = 무슨 일이에요? = 무슨 일이세요? = 무슨 일입니까? = 무슨 일이십니까? all basically mean the same thing. Only the difference is level of politeness and formalness. Just one thing - '무슨 일이요?' is not really used in reality nowadays, because it sounds like very old-style Korean, and it sounds less polity than 무슨 일이에요 even though it sounds more polite than 무슨 일이야 because of the honorific ending '요'. The point is it doesn't sound very polite and it sounds very old-fashioned, sounds like something used 500 years ago.
**So yes basically , 이야, 이에요/예요 and 입니다 all stem from the base form 이다. The difference between usage of 이에요 and 예요 is if a noun or a verb is ending in a consonant (3 letter word), it is conjugated using 이에요 and if a noun or a verb ends in a vowel (2 letter word), it is conjugated using 예요. Example "I am Indian" - 저는 인도 사람이에요 (사람 ends in consonant ㅁ so 사람 is conjugated with 이에요). Now if i say "I am an indian woman" - 저는 인도 여자예요 ( 여자 = woman and since 여자 ends with a vowel "aaaa"/아 ( two letter 자 ), its conjugated using 예요). Now if you are speaking with someone who you know only a little bit or not really that well but its in a casual setting and you want to say " i am an indian woman" as a part of your introduction to someone casually (not too formal) but you still want to sound polite since you and the listener are still somewhat of a stranger for each other, then in that case, you would say "저는 인도 여자예요" but if you are presenting something in a class to your colleagues (presentation etc.), you would say "저는 인도 여자입니다" - strangely, here the vowel rule disappears as far as 입니다 is concerned, a webpage on koreanclass101 explains that ㅂ니다 is only attached to an actual verb so regardless of whether your name ends in a consonant or vowel, 입니다 is used to conjugate your name Or i guess any noun like "여자" for that matter. So basically "I am Ruchi = 저는 루치예요 = 저는 루치입니다" - Hope this helps.
ㅂ니다 vs 입니다
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