What DarkWanderer wrote is correct, the rules for critical success or failures apply to where they are stated such as attack rolls in combat.
There has always been room in D&D for house rules and variations. In the DMG on p.239, the Automatic Success variation is discussed which is up to DM discretion, so long as the judgement is consistent and predictable. All D&D versions have given the DM some flexibility and their judgement to run a fun game (p.236 of the DMG, for example, which talks about using or ignoring the dice, or Inspiration a couple of pages from there).
You could play a House Rule that says 1 fizzles or fails. (http://forum.rpg.net/showthread.php?740888-5e-Your-house-rules is a thread discussing house rules and variations others have adopted in their own games.)
Since AD&D 1st ed, I have typically played 20 as success and 1 as failure no matter what, even for casting spells that require a die roll to hit (eg. Witch Bolt, PHB p.289 requires a ranged attack roll). A 1 is a critical failure, which allows some descriptive effects about fizzling or fumbling:
- "The air crackles with electricity then poof - the smell of sulfer and black soot surround the Wizard's surprised face as the spell fizzles!"
- "Before the last syllable rolls off Dramomere's tongue, he fumbles his components and drops them on the floor! The spell fizzles!'
Critical failures can be fun, everything from a character tripping and going prone to clubbing their party member or sticking a rapier through their own shoe, and use the opportunity for extra descriptiveness (likewise with critical hits).
However, these are all variations and not core 5e rules.