Short answers, pulled from the story.
Roman assemblies were formal meetings of the Roman people convened by a magistrate. They elected all magistrates during the Roman Republic, enacted legislation, decided questions of war and peace, and determined guilt in legal proceedings.
A comitia was a voting assembly where citizens were organized into blocks to make decisions on elections, laws, war, or guilt. A contio was a non-voting gathering where citizens listened to speeches without enacting anything.
Roman assemblies used three kinds of voting blocks: curiae, centuriae, and tribus. These gave rise respectively to the curiate, centuriate, and tribal assemblies. By the middle and late republic, only the centuriate and tribal assemblies remained politically relevant.
The curiate and centuriate assemblies date to the regal period, before the republic existed. After the overthrow of the Roman monarchy, traditionally dated to 509 BC, the centuriate assembly is said to have elected the magistrates who became the consuls.
The Roman assemblies had become entirely obsolete by the third century AD. Their decline began when emperors accrued direct legislative power and the senate became the de facto electoral assembly after the fall of the republic.
Ordinary citizens had no presumptive right to participate by speaking in Roman assemblies. Only speakers invited by the presiding magistrate could address the crowd; citizens were present to listen and then vote.