Which war was most important in establishing rome as the greatest empire in the world?

The correct answer is the Second Punic war

Basically, after the First Punic War, Carthage which controlled much of the Mediterranean area was weakened a lot but still kept its land and status. The Romans wanted to end them once and for all and take their territory so they started the Second Punic war and managed to completely take Carthage and dominate the entire Mediterranean area.