Howdy! Thanks for asking a neat question!

So to start, let us talk a bit about the middle to late Carthaginian military. Polybius states that Carthage's military consisted entirely of mercenaries, but how true is this?

The statement itself stems from Carthage's lack of citizen soldiers; in a time period when citizen armies (see Greece, Rome) were the norm, Carthage was notoriously hesitant in using its population for war. And why wouldn't they be? A series of devastating military defeats had caused massive loss of life to the Carthaginian citizenry in the past (see Himera, Krimissus, and others). To add to this, the Carthaginians were wealthy and powerful enough to get others to fight wars anyways.

Anyways, this peculiarity about Carthaginian armies would have stuck out to the ancients, and that is why Polybius takes note of it. But in reality, the characterization of the army as an army of solely paid mercenaries does not quite fit. So who was in the army then?

First, there were lots of Libyans. Most of the Libyans in the army were not full Carthaginian citizens, but were from cities and towns which had made treaties with Carthage to provide money and soldiers. Someone with a keen eye might find parallels to the Roman system. The Libyans might have been paid some amount, but for the most part, this was not what kept them in service; they were essentially conscripted.

Phoenicians seem also to have served in the armies, and are mentioned with some frequency. These Phoenicians could be levied from Phoenician cities in Africa, or alternatively, from the warzone where Carthage was operating (namely Sicily). Similar to the Libyans, the Phoenicians were probably not in it for money either, but as part of a treaty, by request of Carthage, or simply because of mutual interest. Since Carthage was the protector of the Phoenician cities and people in the West Mediterranean, it was in most of the Phoenician cities' best interests for Carthage to win wars.

For cavalry, Carthage would frequently turn to its neighbors in the West, the Numidians. The Carthaginians were less dominant over Numidia than they were Libya, and so we should assume that sometimes the Numidians served by treaty, and other times for pay.

What's that you say? Jollydevil, Polybius can't be totally wrong! And he's not. There were a LOT of mercenaries in the Carthaginian armies, especially later in Carthage's lifespan. What is of particular interest is where they came from. We have mentions of mercenaries from Gaul, Spain, Italy, Sicily, and a host of other places. In fact, there are theories that the Gauls who sacked Rome in 390 BCE may have been making their way to Sicily to fight as mercenaries for Carthage.

Occasionally, Carthaginian citizens would serve in the military, particularly earlier in Carthage's lifespan. In times of crisis though, Carthage was not afraid to levy citizen troops to fight its wars. This happened during the invasion of Agathocles, the Mercenary War, and each of the Punic Wars, to name a few instances. That said, the Carthaginian citizenry had a very poor service record and their conscription was usually a last resort.

Finally, we have individual recruits: these might be ex-slaves (Spendius in the mercenary war was an ex-slave, and Carthaginians seemed to have few qualms about freeing slaves for service in the army), or individuals seeking a new profession. There would have certainly been officers of Carthaginian descent in the army too, particularly in the upper echelons. Carthage was a very diverse place, and one probably would have found a number of part Punic, part Sicilian/Libyan/Italian/etc. people serving in the armies.

And how about the navy? Ancient sailors were generally drawn from the lower levels of society, as well as from island areas; Carthage almost definitely pulled sailors from both of these areas. It is helpful to remember here that fleets in ancient times were immensely expensive, not only in money but in manpower as well. A fleet of 200 ships would have required somewhere around 40,000 people to fully man. Thus, we have to assume that Carthage pulled sailors from anywhere it could. The sailors under direct Carthaginian service were not mercenaries, however, Carthage also employed a large number of privateers, who we might consider closer to mercenaries of the seas.

OK! So we've got some good info: how would Carthage actually have put together an army? It would have varied from situation to situation, but there would have been a call for mercenaries first and foremost; mercenaries could come from far away, but they could also come from relatively close by. For instance, we know of a couple of Greek officers in the Punic army, 212-211, who lived in the city of Carthage and were descendants of a Sicilian Greek migrant.

Calls for conscription would have gone out as well, with messages being sent that such and so city was to provide the state with however many men. If a general happened to be serving in, say, Spain when war broke out, one would expect to see more Spanish conscripts in the army than other peoples.

The initial call was not the end of recruitment though; the Carthaginian army would have kept recruiting at (and on its way to) the front. For instance, Hannibal recruited Gauls en-mass into his army while descending into Italy.

Hope this answers your question!

Sources:

Polybius, Diodorus Siculus, Polyaenus

Carthage Must be Destroyed, Karthago: Studien zu Militar, Staat, und Gesellschaft