In the republican period most of the Roman magistrates were elected

Consuls: 2 Roman magistrates with imperium

The most important magistrates were the military commanders, the 2 consuls

Dictator: a powerful Roman magistrate

In times of great crisis one of the consuls could appoint a dictator. A dictator in Roman context should not be confused with modern day dictators, but likewise he had absolute power. He was to stay in power until the problem was solved or a period of six months had lapsed. When none of the consuls could be in Rome for the elections, the senate would appoint a dictator to organize the elections. The dictator would lay down his powers immediately after the elections.

       

The dictator  was to go on foot to  command the infantry.  But when the over 60 years old dictator Q. Fabius Maximus asked permission to ride a horse, the senate granted him this privilege.
The dictator then appointed a magister equitem  (master of the horse)  to command the cavalry.
The people and the senate proved not to be rendered completely powerless though. When in the hannibalic war the dictator Q. Fabius Maximus refused to attack Hannibal, the senate empowered his magister equitem M. Minucius Rufus to be equal in rank as Fabius. As expected Rufus attacked Hannibal with his half of the army but lost.

According to Dionysius of Halicarnassus(5.70) the first dictatorship was created to subordinate the Roman plebeians, who made problems about their debts.

Censor, this Roman magistrate established the taxes to be paid

The two censors were elected every five years and stayed in office for 18 months. When this office was instated they stayed in office for 5 years; the period between two cencuses. In 434 bc already this was shortened by Mam. Aemilius in the time he was dictator. Their main task was to conduct a census of the Roman people. This meant taking statements under oath from all male citizens, about their property. On this statement they assigned that citizen and his family to a place in the comitia centuriata, to a tribe and to an economic . Also the amount of tax to be paid was established on basis of this property.

These Roman magistrates first appeared  in 443 BC. From then on there were sensei held on a irregular basis till late in the 4th century, when the 5 year cycle was fixed.
At the end of a census was held a purification festival called Lustrum. The first time a census was held, was traditionally attributed to king Servius Tullius. T. Livius (1.44):

After the whole army had been drawn up there, he purified it by the triple sacrifice of a swine, a sheep, and an ox.

This was called "a closed lustrum," because with it the census was completed.

Another task of the censor was to guard the moral of the citizens,especially the senators. He would judge them (un)worthy to serve in the senate.
Because there were 2 censors, this could lead to problems if the 2 had problems with each other (See Livius 29.37)

Quaestor the lowest Roman magistrate

The first time the office of quaestor shows up is in Livius' Ab urbe condita 3.24 in the story of Kaeso Quinctius 458BC. According to Tacitus the office of quaestor was first recreated in 446. This was the lowest of the Roman magistrates and a good starting point for a political  carrier. Initially their job was to take care of the spoils of war. They served also as assistant for the consuls. As to be seen in the text of Tacitus' Annals 7 the number of

The quaestors indeed were appointed while the kings still ruled, and this the revival by Brutus of the lex curiata plainly shows. The consuls retained the power of selecting them, till the people bestowed this office as well as others. The first so created were Valerius Potitus and Aemilius Mamercus sixty-three years after the expulsion of the Tarquinii, and they were to be attached to the war-department. As the public business increased, two more were appointed to attend to affairs at Rome. This number was again doubled, when to the contributions of Italy was added the tribute of the  provinces. Subsequently Sulla, by one of his laws, provided that twenty should be elected to fill up the Senate, to which he had entrusted judicial functions. These functions the knights afterwards recovered, but the quaestorship was obtained, without expense, by merit in the candidates or by the good nature of the electors, till at Dolabella's suggestion it was, so to speak, put up to sale.

Military tribunes, high ranking officers in the Roman army

In the period from 405 to 367 BC there were often no consuls but instead mostly 6 military tribunes with consular power. According to Livius this was a kind of compromise between the patricians and the plebeians. The latter demanded access to the office of consul. This new office was open  for plebeians. These tribunes were in fact military tribunes who served as high ranking officers in the Roman army, but now a number of them had consular emporium (power).

Praetor, one of the highest ranking Roman magistrates

The office of praetor resembled in many ways that of consul. Possibly the function of consul had been called praetor at the beginning of the republic. In 367 BC this office was created (again?) together with the office of curule aedile. Both the positions the patricians demanded for themselves. T. Livius suggests that this was done to compensate the patricians for the loss of one post of consul to the plebeians. The praetor acted as chief judge and deputy of the consuls. Within three decades the first plebeian would hold the office. When the roman wars became more complex and further away from home, praetors (and propraetors) would be appointed as military commanders.

(Curule)Aediles, Roman magistrate to organize the games

Aediles were to supervise public works like temples. Their main job however was the organization of the games. Though this function was good to gain popularity, it would also cost a lot of money because a lot had to be paid out of the private means of the aedile. Aedile had been a plebeian function until the function of consul was opened to the plebeians in 367 bc. Curule aedile was created for the same reason as praetor and intended only for the patricians.

Plebeian tribune, a Roman magistrate to control consular power

Next to the consuls was the office of the plebeian (or peoples) tribunes (tribuni plebis) the most important. They could propose laws and so they did frequently. Like consuls they were equal in power and could block (Veto= I forbid) the proposals of the senate , consuls, or even their colleagues. This rule, though it was designed to protect the people against rash decisions of the senate or consuls, would be exploited by the patricians against the plebeians.   They could prosecute any curule magistrate before the people after the term of that magistrate had ended. They were not allowed to leave Rome for more than a day. But also here the Romans would prove to be very flexible if needed. When the senate had doubts about the discipline P.Cornelius Scipio maintained in his troops on Sicily, a committee,with among others 2 Peoples tribunes, was sent to inspect these troops. This journey will have cost them close to a month time! (see Livius 29.20)