Learn French: Il ignorait la plainte du voisin depuis des mois. - He had been ignoring the neighbor's complaint for months.

In French, when an action started in the past and was still ongoing at a specific past moment, you use the imparfait with depuis. English uses the past perfect continuous ("had been doing"). Just like the present tense plus depuis means an action is still going on now, the imparfait plus depuis means it was still going on then.

Could I use the plus-que-parfait (il avait ignoré) instead to mean "he had been ignoring"?

No, because the plus-que-parfait implies the action was entirely completed before another past event ("he had ignored"). Since he was still actively ignoring the complaint at the time we are talking about in the past narrative, you must use the imparfait with depuis.

I thought ignorer meant "to not know". Can it also mean "to ignore"?

Yes! While ignorer is a famous false friend that often means "to be unaware of", it can also mean "to ignore" when referring to intentionally disregarding a rule, a warning, or a complaint like la plainte. Note that if you are deliberately ignoring a person, it is usually more natural to use ne pas faire attention à.

Why do we use depuis here instead of pour if the English says "for months"?

Pour is primarily used for future intended durations, such as going on vacation for a week. When an action has been going on for a period of time up to a certain point in the present or past, French always uses depuis, even when English translates it as "for".