The answer your book doesn't want
In English, as in many languages, nouns change form:
This is called inflection. In the example above, rat has two forms. These forms reflect grammatical number (singular and plural), so we can say nouns inflect for number.
But some words inflect for other purposes. In English, for example, we find pronoun forms like these:
How are these forms used? Well, an exact description is pretty complicated! But the main distinction we find is between Subject and Object function:
I punched Sally. (pronoun in Subject function)
Sally punched me. (pronoun in Object function)
And when we find a class of words with this sort of distinction, we call it case. Labels like "nominative" and "accusative" describe inflectional forms whose main function is Subject or Object.
Of course, that doesn't mean it's their only function:
Q: Who wants ice cream?
A: Me! (accusative form, but not in Object function)
But the labels are still useful as long as it's their main function.
Of course, the exact label isn't that important. Some people say "subjective" instead of "nominative", because they think that more clearly communicates the idea of Subject function, and they do likewise for "objective" instead of "accusative". But let's go with nominative and accusative for now, because those are the labels you used in your question, and because they're commonly used by linguists.
So we've decided that pronouns have accusative and nominative cases. What about common nouns? Let's take a look:
The rat bit me. (noun in Subject function)
I bit the rat. (noun in Object function)
We find the same form, rat, in both functions! There doesn't seem to be a contrast in form, so we can't really say it has inflections whose primary functions are Subject and Object. And in fact, no matter what common noun we choose, we'll find the same thing. The nominative-accusative contrast simply isn't relevant to this word class!
I mean, sure. If we really wanted to, we could pretend it inflects like this:
rat (nominative neuter singular)
rat (nominative masculine singular)
rat (nominative feminine singular)
rat (accusative neuter singular)
rat (accusative masculine singular)
rat (accusative feminine singular)
rats (nominative neuter plural)
rats (nominative masculine plural)
rats (nominative feminine plural)
rats (accusative neuter plural)
rats (accusative masculine plural)
rats (accusative feminine plural)
But we're just making our life harder! Why make things more complicated when there's no benefit to doing so?
I bet you think it's silly to invent a gender system for rat, right? The word never changes form to reflect a neuter/masculine/feminine contrast, so why would we make something like that up? We can say the exact same thing about the accusative-nominative contrast. No common noun changes form to reflect gender, and no common noun changes form to reflect Subject or Object function.
So let's make our lives easy. We'll say rat can't bear accusative or nominative case. Pretending it can doesn't tell us anything useful about the language.
What about that? Well, traditionally it's considered a relative pronoun, like who. Let's pretend that's correct for a moment and look at that:
It was Sally [ that ＿＿＿ punched me ]. (gap in Subject function)
It was Sally [ that I punched ＿＿＿ ]. (gap in Object function)
Does that change form depending on whether the gap is in Subject or Object position? It does not! In fact, that has no inflections in Standard English. Does calling it accusative or nominative tell us anything useful? It does not.
But wait, is that really a relative pronoun? In A Modern English Grammar on Historical Principles, the great Otto Jespersen suggested that it makes more sense to consider relative that the same marker of subordination we find in non-relative subordinate clauses. And although it's not entirely uncontroversial, most modern linguists follow Jespersen here, including Huddleston, Pullum, and Peterson in The Cambridge Grammar of the English Language (p.1056) and McCawley in The Syntactic Phenomena of English, 2nd ed. (p.429).
The arguments in favor of this analysis are complex, though, so I'll omit them here; if you'd like to know more, please ask a separate question about whether relative that is a relative pronoun!
But assuming for a moment that relative that isn't a pronoun of any kind, it makes even less sense to say that it bears accusative or nominative case! Why? At least with pronouns, we can suggest that other words in the same class bear the contrast, and it makes sense to set up a system of case for an entire class, even if it doesn't show up for every word in that class. But no English subordinator inflects under any circumstance, so we have no justification for claiming that belongs to a system of case, even indirectly.
So neither rat nor that bears case here. Neither word is accusative or nominative. Your book's definition is not particularly useful in terms of description, so I think we can safely ignore it.
The answer your book wants
If you really have to use the book's definitions because of school requirements, just look at the function of each part of the sentence:
The cat killed the rat [ that ＿＿＿ ate the corn ].
The rat is an Object, and therefore the authors believe the noun rat is in "accusative form".
The cat killed the rat [ that ＿＿＿ ate the corn ].
In the relative clause, the gap is in Subject position, but the authors of the book are still living in the land of pre-Jespersen grammar, and they believe that is in Subject position:
The cat killed the rat [ that ate the corn ].
So they think that is a Subject, and therefore they believe it's in "nominative form".