What is the difference between the forms of "and" "и" and "a"?
I could not really recall any example when Russian
a can be translated as
and in English. Usually it is
but in English. But then I looked into a dictionary and there was such a meaning.
1) и (сочинительный союз, реализующий различные логические отношения между соединяемыми частями)
boys and girls — мальчики и девочки
2) а, но (противительный союз)
I shall go and you stay here. — Я пойду, а ты оставайся здесь.
There are books and books. — Есть книги и книги.
The difference is that the former is a coordinative one and the latter is an adversative conjunction. The first connects some words, which are not opposing each other in a particular context. And the second is something like an opposite, it connects words, parts of a sentence which do oppose to each other.
So boys and girls do not oppose, you are sort of enumerating the options.
With the I shall go and you stay here example there are two parts of the sentence which do oppose.
And There are books and books even in the dictionary was translated as
и which also has the same meaning in Russian.
And if I encountered such examples in English I would not use the
a conjunction, really. I would simply translate it as
и in every example of the provided ones, as it was done with the third one.
I believe one good way to conceptualize 'a' in Russian and how to differentiate it from 'и' is the conjunction 'whereas.' In most cases I believe it is a good approximation, noting that 'whereas' generally means 'compared with the fact that; but.' When there is a clear, easily divisible contrast between the first and the second clauses of a sentence, it sets you down the right path to using it properly yourself when found in other contexts.
Certainly it doesn't cover all cases. In English, when it comes to bringing two clauses together there are a lot of subordinating conjunctions (after, although, as, as if, because, before, even if, even though, if, if only, rather than, since, that, though, unless, until, when, where, whereas, wherever, whether, which, while). As an English speaker you've internalized how to use these words largely unwittingly. If you wanted to explain when you use any of these and how one differs from the others, I think it would require a concerted mental effort to do so. Think analogously that Russian speakers have the same concepts, but some of which are wrapped up in 'а' but cannot be consistently translated as the same word in the English. It requires context to see its variety of meaning and you'll gain that with exposure, like you did with English.
I'll give you a couple examples to see how it varies, even when among sentences where it seems to follow the same structure:
Звонок брякнул слабо, как будто был сделан из жести, а не из меди.
- The bell rang weakly, as if it was made out of tin and not out of copper. (/as though it were made not of copper but of tin.)
На ее тонкой и длинной шее, похожей на куриную ногу, было наверчено какое-то фланелевое тряпье, а на плечах, несмотря на жару, болталась вся истрепанная и пожелтелая меховая кацавейка.
- On her thin and long (we would say 'long, thin' in English) neck, which resembled a chicken's leg, was wrapped in some kind of flannel rag, whereas on (from) [her] shoulders, despite the heat, dangled an all (entirely) worn-out and yellowed fur katsaveyka (jacket).