Magic attacks, also called Ranged Attacks in some of the game's text, are one of the two primary ways enemies deal damage to the player in the Battle Arena (the other way being through Melee Attacks). They are typically much weaker than Melee Attacks, but can still be very debilitating in certain circumstances.
What Monsters have Magic Attacks
In general, you can find the answer to this in your Bestiary; the stat M. Attack gives the power of the enemy's Magic Attack. If an enemy has 0, they have no Magic Attack. In Hardcore Mode, all enemies are given Magic Attacks equivalent to 80% of their standard Attack stat.
What Triggers Magic Attacks
Enemies can only launch Magic Attacks when their HP is below their maximum HP. The rate at which Magic Attacks come is determined only by the ratio between HP and maximum HP. The speed varies between 1 every 21 frames (roughly two a second) and 1 every 70 frames (roughly 1.75 a second). The speed changes by 1 frame every 2% damage, so a monster at 50% health will fire magic attacks at 70 - 25 = 45 frames per attack (1.125 attacks per second).
Receiving Magic Attacks
Magic attacks do not ignore Damage Resist--in fact, they are reduced by both standard defense and Magic Resistance. They can not be absorbed using the Absorb skill, but can be absorbed with the Magic Reflect skill. Magic Attacks can be dodged, just like a Melee Attack, using either Dodge Chance or Evasion.
If you have Magic Reflect, it will trigger against every magic attack, even ones that are absorbed or dodged.
Getting hit (as in, failing to absorb or dodge) a Magic Attack will cause the player to lose half their combo, up to a maximum of 200.
Like Melee Attacks, if your enemy has a special ability, it will have a 7% chance of triggering if your enemy successfully hits you with a Magic Attack (in other words, if you fail to absorb or dodge it). If your enemy has an ability with multiple components (Like Chaos), each individual component will have a 7% chance to trigger, with the possibility that many will trigger at once.