Missiles is a common name for projectiles from ranged attacks. Note that missiles are not a synonym to ranged attacks.
In Diablo I, most ranged attacks and spells are projectiles, including arrows. They need to directly hit their target in order to have any effect, therefore casting a spell or making a shot also requires some degree of accuracy, especially against moving targets. Most obstacles, including allies, between the player and the target, will block any and all missiles (bar grates being a notable exception). Monsters, however, can shoot through allies without harming them.
In Diablo II, Missiles are any non-magical (that is, physical) projectiles, including arrows, bolts, and thrown weapons. This term is used in reference to Missile Weapons, and when an item has a Defense vs. Missile bonus. Magical missiles bypass this extra protection.
In Diablo III, any solid projectile (regardless of damage type) that has a distinct speed and linear trajectory counts as a missile. Their main distinction from other ranged attacks is the fact that once released, the missile flies forward (some can home on the target if needed, or otherwise change trajectory), hitting the first enemy target that happens to be in its way. Hitting a solid wall usually causes missiles to stop immediately, and allies are ignored for this purpose (missiles pass through them harmlessly).
Some missiles are piercing, and once they hit the first target, they continue to move in the same direction, potentially hitting more than one enemy standing in a line. Some of them can even bypass obstacles. Other missiles explode on impact, disappearing and hitting all enemies within a certain radius.
Some skills, like Arcane Torrent, Firebats or Cluster Arrow, are not missiles, even though graphically they may be depicted as such. Such skills are dubbed torrents, as they create a stream that reaches the destination point, bypassing most or all types of obstacles in its path. They are not affected by special effects that alter the missiles' behavior. Even if they create just one single blast, they are still not missiles. Same goes for Mortar monster affix and monster attacks like Lacuni Huntress grenades or Subjugator firebombs.
Skills like Disintegrate, Fires of Heaven or Ray of Frost also do not count as missiles, and are instead dubbed beams. They reach their target instantly, but stop if they hit a solid wall, and normally hit the first enemy in their path. Piercing beams hit all enemies in line, instantly reaching their maximum length.