There is often confusion when the anti magic field spell is used, armor of invulnerability is worn, or the Beholder's anti-magic ray is employed, whether a given character's ability will still function. The spell does give you some guidelines on how to adjudicate the effects of the loss of magic, but says very little about how to determine what is affected. In short, how do I tell if an ability is magical?
Until recently, the answer has been very much the province of the DM's judgement. Some things are very obvious, like a fireball or a magic missile. Somethings are not, like ki-powered effects or a dragon's flight.
WIzards of the Coast has released a very easy checklist to determine the difference, included in the Sage Advice Compendium of official rulings.
Within the worlds of Dungeons and Dragons, there are two types of magic:
the background magic that is part of the D&D multiverse’s physics and the physiology of many D&D creatures
the concentrated magical energy that is contained in a magic item or channeled to create a spell or other focused magical effect
Effects that cancel, dispell or nullify magic are concerned only with the second kind of magic. The first is just assumed to be part of the natural physical laws that allow a fantasy world to exist. But again, how to tell the difference? The document gives us a test:
Is it a magic item?
Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
Is it a spell attack?
Is it fueled by the use of spell slots?
Does its description say it’s magical?
If any one of these questions can be answer "yes", then the effect, ability or item is magical for the purposes of being affected by magic cancelling effects.
To use the examples given above:
Fireball: Yes, it is explicitly a spell.
Magic Missile: Yes, it is explicitly a spell.
Dragon flight: No. It is not a magic item, a spell, or duplicate the action of a spell. It is not a spell attack, or fueled by spell slots. And finally, nowhere in its description is the word "magic" or "magical" used.
Ki powers: Sometimes. Although ki itself is not described as magical, some ki powered abilities use the word magical in their description. So although you could use ki to empower flurry of blows within an antimagic field, you could not use Ki Empowered Strikes to any effect since the description says "your unarmed strikes count as magical" (this is confirmed by Dragon Talk Sage Advice podcast of 10/19/17 at the 29:00 mark)
A few more commonly encountered problem abilities:
Paladin's Divine Strikes: Yes, they are fueled by the expenditure of spell slots. Improved Divine Strike appears to be non-magical however, since it fails to meet any of the criteria. Perhaps the paladin is now so pure and devoted to his oath, the he or she is tapping into that other kind of magic. (Devoted question)
The Portent and Third Eye abilities of the Diviner Wizard: No, they do not meet any of the criteria. It must be assumed that they are natural abilities based on a heightened level of awareness possible within a fantasy universe.