The Roman world didn't really have a separate judiciary. Such an idea, at least in Europe and as we understand it today, is much later. Courts and trials were folded up into other institutions. This was true at Rome as it was in all the states of the ancient Mediterranean. In particular, justice was the responsibility first and foremost of the magistrates.

In the very earliest period, the magistrates--there being at this time only one magistracy, the consulship--heard cases and dispensed justice. Soon, two of the praetors took over this responsibility, the urban praetor and the peregrine praetor dealing with cases involving only citizens and involving citizens and foreigners, respectively. This basic arrangement held true for Italy throughout the entirety of Roman history, with these two praetors effectively overseeing the legal system, particularly the system of civil law. Roman law differentiated between civil law (ius civile) and criminal law (for which there isn't really a single Latin word). Civil law took up the bulk of the praetors' duties. Each year, the most significant magistracies issued edicts laying out how they intended to carry out their duties of office. While for the most part these edicts were tralatician (i.e. they carried over in bulk from one year to the next), the urban praetors typically made rather substantial changes, setting out new classes of action and effectively creating a whole bunch of rules and regulations that were not, formally speaking, governed by law but had the same effect. These changes (ius honorarium) were basically the result of things that the praetors had had to deal with, and from 67 on they were also binding on the praetors who issued them themselves, thanks to a Lex Cornelia. The execution of civil law is relatively straightforward, but it rests quite a lot on the praetor's edict. At Rome, regardless of whether an offense was covered by civil or criminal law, no action occurred until a private party brought suit. In cases of civil law, suit was brought before either the peregrine or the urban praetor, depending on the circumstances, and the praetor would make a ruling based on the hearing, which might consist of an actio in factum or other action on the spot if things were pretty clear-cut, or could consist of arrangement for a formal trial with a judge (iudex).

Criminal cases were rather less systematic. Typically a criminal case was also a political one, e.g. treason, so they were seen as a matter not for a closed court but for the entire community, since they were crimes against the citizen body general. Whereas civil cases were tried by a single judge, criminal cases were tried by a group of judges that rendered their decisions by way of secret ballot. The size of these juries varied, but in the permanent courts they were as large as seventy-five men (because the judges were always men). Importantly, the Romans described criminal trials as "quaestiones," which is to say "inquiries." The distinction between court as a place where justice is heard and dispensed and a place of investigation in Roman criminal law is very fine. The Romans had no investigative police force, so it was the duty of the two parties to provide the necessary evidence and show by argumentation that it was significant and valid--witness testimony was of course very important.

Were these judicial actions overseen by professionals? For the most part, technically no. The praetors were career politicians first and foremost, and while they needed a relatively good knowledge of law (and had likely cut their teeth on forensic oratory at the beginning of their careers) as part of that role, they were not professional jurists in the way that a British barrister or the French magistrats are. They were, however, assisted by professional jurists, a somewhat shadowy group that developed slowly over the last couple centuries of the Republic and really only became fully developed during the imperial period, when civil and praetorian law reached their greatest maturity. These professional jurists assisted the praetors in coming to their decisions, and they helped them write the edict. Or, more accurately, they wrote the edict for them based on their orders and working alongside them, since it was the professional jurists (and, in particular, the scribes) who actually had the technical knowledge to compose legal documents. While Rome had at best only the most minimal professional bureaucracy, even in the imperial period but especially during the Republic, we should not imagine the praetors sitting alone puzzling through reams of imposing legal documents, but rather understand that they were aided in their operations by dozens of helpers, ranging from friends and colleagues to the scribes provided by the state, whose technical knowledge contributed to produce a unified whole.

Then, what about the judges? Judges were, by definition, legal amateurs, as technically were the magistrates. Iudices were private persons, but their role was to decide on the case, not to preside over it--the actual bit in civil suits before the iudex was pretty informal. In reality, it appears that iudices tended to have substantial legal knowledge and were, at least by the imperial period, also professional jurists or semi-professionals. Gellius, for example, recounts his first experience as a judge, as well as how he asked the legally-inclined philosopher Favorinus for his opinion. Occasionally some civil trials were judged by other bodies, such as the board of centemvirs, who were drawn from other bodies (notably ex-quaestors, which gives a sense of what class of people judges came from). Iudices, both in civil and criminal suits, were chosen from a pool of eligible men, and they were chosen by the parties. In criminal quaestiones this was rather complicated, and depended on the court. Initially, quaestiones were set up by special action, typically a decree of the senate but sometimes a law of the assembly, and occasionally this continued into the late Republic (e.g. the assembly's ratification of the law that established a trial for P. Clodius' sacrilege at the Bona Dea in 62, despite the protests of the promulgating consul). The selection of judges in these cases was not necessarily fixed, and depended on the rules that the senate or the people had determined. Indeed, these early special investigations didn't really have jurors at all. Rather, the magistrate in charge would call a council (concilium) of people of his choosing to help him deliberate over the case and render judgment.

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Judges for most "court cases," as they would be conceptualized in modern times, were administered by a praetor. This was an elected magistrate's position which, by the Middle Republic, was subordinate to the two consuls. Men typically ran for the office of praetor after first serving as quaestor and then possibly aedilis. Praetors held imperium, the power to make binding decisions, and they could be assigned military command roles when needed, even after their term (prorogated with imperium pro praetore). The praetors did not have sweeping powers, but were typically assigned to specific functions or tasks, including specific judicial spheres. They could also lead special investigations, head up committees, or prosecute a specific military objective in the place of a consul.

In the earliest days of the Republic, there was only one (possibly three?) praetor, and there is a lot of confusion about the origins of the position in relation to the consuls. The praetors in early days were called "colleagues" of the consuls, and Livy tells us that they were even back then meant to deal with legal concerns (*ius in urbe dicere, 6.42). By 246 BCE there was a second praetor in operation, the praetor peregrinus, who's job specifically dealt with matters between Roman citizens and non-citizens within the city. This differentiated the "original" praetor as the praetor urbanus. By 197 BCE, four more praetor slots had been created, and we are told it was mostly to help govern newly-organized territories in Sardinia, Sicily, and the Spains, mostly in their typical role as judges but also as military commanders, when the needed. Sulla increased the number of praetors per year to eight, and by the end of Caesar's time it had increased to sixteen.

There were no qualifications except to be old enough, and to have been a quaestor previously. You also had to be liked by the electors or rich enough to bribe them. Even in a field supplying sixteen yearly slots, there were a lot of potential candidates drawn from the much larger pool of ex-quaestors and ex-aediles. There was no formalized training whatsoever, though it is assumed that virtually all Romans who were climbing the ladder of offices (the cursus honorum) had extensive informal training under older tutors. The better or more prestigious the tutor, the better prepared the apprentice would be. Cicero was fortunate enough to study under the venerable Q. Mucius Scaevola, along with his colleague and friend Sulpicius Rufus. Certainly this connection furnished a lot of weight to Cicero's candidacy for the praetorship.

On this particular topic, there is an absolute beast of a tome, in two volumes, by T. Correy Brennan: The praetorship in the Roman Republic (Oxford 2000), a full one thousand pages with every imaginable detail of the Roman praetorship included, two appendices, and the kitchen sink.