I ask this question in all honesty.
According to my basic understanding of Italian, a cappella means “in the style of the chapel.”
According to some of my students, a cappella might mean “Like Pitch Perfect.”
Most musicians might say that the best definition is “Without instrumental accompaniment.” But I don’t think that’s quite right, either.
To me, as a composer and arranger, the most substantial difference is that I approach writing for voices differently if they won’t have accompaniment. The way we voice chords, balance timbres, determine range and even key: all these things can change if we’re writing with or without accompaniment.
Fundamentally, it’s the approach of the writing that makes something a cappella. There can be pieces written with no accompaniment that are not written in an a cappella way. And conversely, a piece can have instrumental accompaniment and still be a cappella. For (pertinent to me at the moment) example, an SATB choir with a clarinet part is almost definitely still an a cappella piece because the vocal writing is in the a cappella style. The same is true for a clarinet choir: it’s not without instruments, but it’s definitely a cappella in style.
What does a cappella mean when you use it?
A postscript on pop a cappella: based on my personal definitions, sometimes a cappella is not a cappella. For example, much of Pentatonix’s output doesn’t feel a cappella because of its instrumental approach to bass and percussion. There’s basically always a rhythm section, so the rest of the voices aren’t really written in an a cappella style by my definition.