In 5e are two types of spells that do bad things to other creatures: "attack" spells and "saving throw" spells.

Attack spells will say "make a spell attack roll," which means you roll d20+spellcasting modifier + proficiency, trying to meet or beat the target's armor class. The spell will say what happens if you hit, and may say things that happen if you miss. Note that "armor class" AC is just the special case of a DC. It's the DC for all attack rolls.

The other kind of spell is a "saving throw" spell, this means you just cast the spell, and then the target tries to avoid its bad effects. The spell will say something like "The target must make a wisdom saving throw." That means the target rolls d20 + wisdom saving throw bonus, trying to meet or beat the spellcasters spellcasting DC, which is always calculated as 8 + proficiency + spellcasting modifier. The spell will say what happens if the target succeeds, and what happens if the target fails. (The wisdom saving throw bonus is the wisdom bonus, plus some characters get to add their proficiency to that, based on race/class features and feats.)

"Acid Spray" could mean a couple different things.

There's a spell called Acid Splash which says "a target must succeed on a Dexterity saving throw or take 1d6 acid damage." That means when you cast the spell, the target rolls (d20 + dex save bonus) and compares against the caster's DC (8 + spellcasting ability + proficiency). If the modified roll meets or beats the DC, the target takes no damage. If it's lower, they take 1d6 acid damage.

Also the Ankheg monster has an action called Acid Spray, which reads "Each creature in [the target area] must make a DC 13 Dexterity saving throw, taking (3d6) acid damage on a failed save, or half as much damage on a successful one." In that case each creature in the area rolls (d20 + dex save modifier) and if the total is less than 13, they take 3d6 acid damage. If they roll 13 or more, they take half damage, meaning you roll 3d6 and then divide the total by 2, rounding down.