There are three general cases:
1. Range Attack
The spell says the caster has to do a Range Attack. In that case you do a roll as if you were doing an attack with a range (or melee) weapon, only you use your Spell Casting (Intelligence, Wisdom, Charisma) ability modifier instead of Strength or Dexterity.
From what I've seen, there is no Saving Throw in this case. This is very similar to attacking a creature with a magical melee or range weapon.
2. Saving Throw (like Breath Weapon)
In this case, the caster automatically hits, as with the regular spells. However, the target may escape the spell partially (50% damage) or altogether (no damage, not charm, not disintegrated, etc.) The spell will mention whether the target can roll a saving throw or not.
The DC that you mentioned is what is used by the target to know whether it is hit or not. The DC increases with time as your character goes up in level and its ability increases and proficiency increases making it harder and harder to save against one of your spells.
Note that a creature can choose to fail a saving throw (unlikely, but it is mentioned in several places.) However, I would rule that an object cannot choose because it has no real intelligence.
From what I've seen all Breath Weapons from Dragons, Gorgons, Dragonborns, etc. work in this way. There is no Attack Roll, only a Saving Throw. Like for spells, the DC for the Saving Throw will be indicated along the Breath Weapon of the creature.
Certain other spells offer no saving throw and do not require a range attack. Notably, the Magic Missiles spell is that way.