"The book is at home."
Translation:Liber domi est.
When using the verb esse (here est) to describe a noun, the noun remains in the nominative form. Domi however is in the locative to specify location (since domus is a lucky noun that can use the locative).
The same goes for most other nouns or adjectives used to describe it (there are a few constructions with esse that take other cases, such as the dative of possession).
Did i miss the lesson where we learned about the different forms of nouns and adjectives? I know DL is trying to teach in a conversational way, but i'm just banging my head on these forms, and usually the answer is a word i haven't seen before. Am i actually going to need to buy a Latin grammar book? Egad! Do they even still sell such things in the 21st century?
If you are able to get to the notes/tips, they do introduce the cases. You can also access them from duome at: https://duome.eu/tips/en/la
They do still Latin grammar books in the 21st century. Some may just be reprints of older books, but some do get updated and revised.
If you are looking for a free resource, I tend to look at Allen and Greenough's Latin Grammar for a quick reference. Here is a link to one page on noun declension rules (there are other pages about declensions as well): http://dcc.dickinson.edu/grammar/latin/rules-noun-declension. They also tend to link to videos by the latintutorial YouTube channel.
Librum (accusative) would be used if there was some sort of action being done to the book. Since we do not have anything being done to the book, we are simply describing it, we use liber (nominative).
It is similar in English how "I am home" is said instead of "me am home", "they are home" instead of "them are home", etc.
Does this help?