I don't like the name "accusative" very much as it doesn't reflect the concept, which puzzles the young brains of the learners.
The basic concept behind the name means that a noun in this form of the accusative case "answers the question of" "кого, что" (whom,what) in Russian.
Example: Я виню (I accuse) Машу (accusative for neutral "Маша") за то, что она не позволила мне набить морду придурку, который к ней приставал.
(I accuse her of not letting me break the face of the dork who was hitting on her)
I'm afraid a native speaker of English would be "lost in cases" at the beginning, because they don't understand how a noun can answer the question "whom".

Instead, I'd name this case "objective case", which can simply denote an object of a phrase.
If we take the general pattern "subject verbs object", the object should be in the accusative(objective) case.
I love Mary: Я люблю Машу (Маша)
I hate Mary: Я ненавижу Машу (Маша)
I will kill my neighbor: Я убью своего соседа (сосед)
I want food: Я хочу еду (еда)
I accuse Bill Clinton of not letting the world watch online him playing with Monica. Я обвиняю Билла Клинтона в том, что...

Objects (underlined) are in the "accusative" form. The nominative (basic) forms are given in the round brackets right after the phrases.