what kind of government did Carthage have?

Answer 1

Carthage was ruled by Suffets (judges). Two judges were elected annually for life, so a large group of judges, which had both legislative and judicial powers, existed at all times. There was a part of the judges' corps called the Hundred and Four that existed solely to oversee the armed forces and judge generals. Polybius mentions a Carthaginian Senate and the fact that generals reported to it, so it is possible he was referring to the Hundred and Four.

Both Eratosthenes and Aristotle mention the Carthaginian constitution.

At this time, it is not known how restrictive Carthaginian electoral law was. Considerable difference of opinion exists on the subject. It is also possible that the definition of eligible voter changed over time.

Answer 2

Carthage appears to have been an oligarchy -- a form of government in which power rests in the hands of a small elite group.

Historians don't have many primary sources left from Carthage. They were mostly destroyed after their defeat by the Romans. What we know of them comes from descriptions written by their contemporaries -- the Romans & Greeks, especially.