My car broke down.
'Guasto' — GUA-sto. The 'ua' is a diphthong — smooth sound. Stress on the first syllable.
Use when your car has broken down and prevented you from getting to your destination. Completely credible and generates immediate practical concern rather than social awkwardness.
'Guasto' means breakdown or fault. 'Ho avuto un guasto' (I had a breakdown) uses 'avere' + noun — different from 'la macchina si è rotta' (the car broke itself) which uses 'essere'. Both are correct; this version is slightly more formal.
La macchina si è rotta.
The car broke down.
More colloquial — 'rompersi' (to break itself) is the everyday version
Ho una gomma a terra.
I have a flat tyre.
'Gomma a terra' (tyre on the ground) — extremely credible and common
Non parte.
It won't start.
Short and desperate-sounding — generates immediate sympathy
'Soccorso stradale' (roadside assistance) is provided in Italy mainly by ACI (Automobile Club d'Italia). Italian drivers have strong brand loyalty to ACI and it's the first call made in a breakdown.