According to Jisho, 病気 is a noun meaning 'illness'. I recently read the sentence Aは病気です, which I took to mean "[Person] A is ill". My confusion lies in whether 病気 is properly an adjective, i.e. "sick/ill", or an abstract noun, i.e. "sickness".
病気 is a noun, which describes a state. Because of this, it can be paired with the copula だ to describe the state of something or someone, just like in English.
A is sick.
です is just the polite form of だ. We could've used either.
Sometimes, it can be used as a "no-adjective", as in the following:
I called on my sick friend everyday.
However, what's really happening is that this の is not a particle but one of the alternant forms of だ, being な the other one. 病気の友人 means "a friend who is sick," which becomes "a sick friend." This doesn't translate as "friend of sickness."
Let's look at another example.
The medicine cured him of his illness.
Now this is the particle の.
Notice the word order: 病気の and の病気
Let's look at jisho.org's entry carefully. It says:
Noun, Noun which may take the genitive case particle 'no'
In other words, 病気 is a no-adjective as well as a noun; it can work like the English adjective "ill".
As I understand it, the term “no-adjective” simply means “nouns which are typically translated to adjectives in English and other languages.” If we treat Japanese as a language in its own right, distinguishing them from nouns as different parts-of-speech is completely artificial.
The term "genitive case" refers to having a noun as modifying another word, usually a noun. That is, having a noun as adjective.
As these articles say, 病気 is still technically a (stative/descriptive) noun. But note that that English has similar noun usages, too, for example "person of credit" (= credible person), "man of courage" (= courageous man). So you may think the normal Japanese way of saying "ill person" happens to be closer to something like "person of illness". Aは病気だ is "A is of illness" rather than "A is illness", just like Aはフランスだ means "A is in France" rather than "A is France".