"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.