Like so many things in 5th edition, if a feature has a special treatment in a specific situation, it will mention so.
Regeneration, for example, may be worded as such:
"Regeneration. The oni regains 10 hit points at the start of its turn if it has at least 1 hit point." (Oni, Monster Manual, pg 239 or here)
or as such:
"Regeneration. The troll regains 10 hit points at the start of its turn. If the troll takes acid or fire damage, this trait doesn't function at the start of the troll's next turn. The troll dies only if it starts its turn with 0 hit points and doesn't regenerate. " (Troll, Monster Manual, pg 291 or here).
You will notice that the spell "Regenerate" (Player's Handbook, pg 271, or here ) does not mention it stops at 0 hp.
When a character has regenerate and he drop to 0hp, does the generation stops? Yes, no? Why?
With every Roll came a new Adventure
Generally no, but depends upon the situation.
If you are talking about the spell "Regenerate", no it doesn't stop. It is like a continuos healing. At 0 hp you are unconscuos regardless.
For the creature, for exmaple the Troll, it regenerates only non-fire and non-acid damage.
Like so many things in 5th edition, if a feature has a special treatment in a specific situation, it will mention so.
Regeneration, for example, may be worded as such:
"Regeneration. The oni regains 10 hit points at the start of its turn if it has at least 1 hit point." (Oni, Monster Manual, pg 239 or here)
or as such:
"Regeneration. The troll regains 10 hit points at the start of its turn. If the troll takes acid or fire damage, this trait doesn't function at the start of the troll's next turn. The troll dies only if
it starts its turn with 0 hit points and doesn't regenerate. " (Troll, Monster Manual, pg 291 or here).
You will notice that the spell "Regenerate" (Player's Handbook, pg 271, or here ) does not mention it stops at 0 hp.