What it seems to be
An invitation to calm down.
What it really is
In Italy, tranquillo is a form of relationship maintenance.
We say it to reassure, to play things down, to take the weight out of a situation. It often arrives before the other person has even had time to worry.
Forgot something?
Tranquillo.
Did someone a favour?
Tranquillo.
Made a mistake?
Tranquillo.
It does not necessarily mean that everything is fine. It means that, at least between the two of us, it is not worth creating tension.
It is a word that protects the relationship before it protects the problem.
Scene
— Sorry, I’m only replying now.
— Tranquillo.
—
— I think I broke the glass.
— Tranquillo.
—
— I still owe you those twenty euros.
— Tranquillo.
The key to understanding it
Italians often try to solve the awkwardness first, and the problem second.