My opinion: "食べて" , "食べろ" and "たべなさい" all means "eat(imperative)". Nuances are different, but , that which is common is far bigger than diffrence. They are all imperatives.

このりんごを食べて。 You can eat this apple. このりんごを食べろ。Eat this apple. このりんごを食べなさい。You should eat this apple.

"食べて" Conjugated verb, but without verbal tense. "食べろ" Verb in infinitive (dictionary form). "たべなさい" "食べなさい" Informal way of saying "please eat"