You are correct. "There are a lot of books" is the correct way to write this sentence. In this case, the phrase "a lot of" is interchangeable with the word "many." I understand why it might be confusing.

If you say 'There is a lot of books' it sounds like you mean literally a lot (a piece of land). It does not make much sense. We say 'There are a lot of books', where the focus is on the books, not on a specific item called 'lot'. 'A lot of' or 'lots of' simply means many. Some may argue that this is not grammatically correct, but I disagree, as we are talking about the books and giving some information about them - we are not talking about a physical 'lot'. Yes, as Teah said above, 'a lot' can be also used in the sense of 'a group of'. In that sense you would use 'there is'. Good point, Teah.

Both are correct, but they mean different things. "Lot" can be used as both a noun and an adjective. A "Lot" (noun) - one group of items (If you have more than one group, you would say "There are four lots of books", for example.) A "Lot" (adjective) - a large number of single items. "There is a lot of books" means that there is one lot (one group) of books. So here you would use the singular "There is". "There are a lot of books" means that there are many single books. You're not counting them as a group here, so you would use the plural "There are".

You are correct in saying you have to use 'there are'. The noun is the key. In your example it is 'books', which is countable, therefore the verb 'to be' must be 'are'. I understand what you are trying to say about the phrase 'a lot of'...it sounds as though it might be singular...but remember you can replace with 'many' or a specific quantity. For example 'There are many books.' of 'There are ten thousand books.' The same is true of similar phrases relating to quantity, so 'There are a large amount of books.' or 'There are a few books.' Only use 'there is' when the noun is uncountable, for example 'There is a lot of milk.' Hope this helps.