"Adesso" and "ora" are synonyms and you can you them interchangeably. However, note that "ora" has two meanings: if considered as an adverb it means "now", but if considered as a noun it means "hour" (singular: l'ora; plural: le ore).
In some regions of central Italy you can also hear the word "mo", that also means "now", but it is only used in informal and dialectal speech, and is never used in standard Italian (even if the famous poet Dante uses it in its Divina Commedia). It probabily comes as an abbreviation of the Latin word "modo", that also means "now".