Meanings
Fig.: to fire someone

Examples
He thought he was an untouchable executive, but after three months in his new position the General Manager gave him the ax for incompetence
Credeva di essere un manager intoccabile, ma dopo 3 mesi nella nuova posizione il Direttore Generale lo ha silurato per incompetenza

The team is doing terribly and rumors have it that one more defeat and the coach will be given the ax
La squadra va molto male e si dice che alla prossima sconfitta l'allenatore verrĂ  silurato

Origin
The origin of this slangy idiom has been related to the axe of an executioner