Player's Handbook, page 194:
Rolling 1 or 20
[...] If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target's AC. In addition, the attack is a critical hit, as explained later in this chapter.
If the d20 roll for an attack is a 1, the attack misses regardless of any modifiers or the target's AC.
Pretty clear, right? If the die says 20, it's a hit and a crit, and I don't care if you have a -38 on the roll. And if you roll an 17 and add +7 to it, you didn't roll a crit even though your total was above 20. Only the number on the die matters.
As to your last line, I'm not sure what you mean about a "natural 1 with positive modifiers". As you can see by the above rule, a natural 1 is a miss no matter what bonuses you could apply to it (unless you have advantage, or some other way to roll again and use the number on the new die instead). D&D 5th Edition does not have any "critical miss/failure/error" rules, so those would be entirely homebrewed by your DM and I can't make any comment on how those would operate.
That said, there's a strong argument for not having critical miss rules. The higher number of attack rolls per turn from highly skilled characters means they'll roll critical misses more often than a low level character would, which seems rather silly.