The elected magistrates in the Roman Republic were held in check by the equal distribution of power through multiple officials of the same rank. The one noted exception to this rule was that of the dictatorship which granted supreme imperium to a single authority. All members of each particular office grouping were of equal rank and could veto acts of other members and higher magistrates (ie Consuls) could veto acts of lower magistrates (ie Quaestors).
As another check on abuse of power, each office was generally a 1 year term with the exception of the Dictatorship which was technically reserved to a 6 month emergency (though this could be extended) and the Censorship (18 months), whose powers were of a managerial nature rather than executive government. The annual term (and varying limits on eligibility for subsequent service) was often a matter of dispute and led to numerous civil disruptions, including the civil war led by Julius Caesar that eventually spelled the end of the Republican system (though it's institutional offices remained throughout the imperial period as well).
Consuls (2) (Latin: those who walk together): The chief civil and military magistrates, elected through the assemblies by popular vote. They convened the senate and curiate and centuriate assemblies. Initially the office was only open to Patricians until the Lex Licinia opened it to Plebeian candidates in 367 BC. According to the Lex Villia annalis passed in 180 BC which established minimum age requirements for all magistrate positions within the Cursus Honorum, Consuls had to be 42 years of age. Under normal circumstances, a Roman could only serve in such a capacity only once every ten years. At the end of their annual term of service, Consuls would take the title Proconsul and generally serve as provincial governors. In the case of the death of a serving Consul, a Suffect Consul would be elected as a replacement for the remainder of his term. They were entitled to 12 Lictors as a symbol of their authority (or imperium).
Praetors (2-8): This magistracy was originally designed as a sort of 3rd Consul and was established in 356 BC for Patricians only after they were forced to share the Consulship with Plebes. This however changed by 337 BC when the first Plebeian Praetor was elected. Romans were eligible to be a Praetor at the age of 39. They had imperium with the main functions being administration of civil law in Rome (Praetor Urbanus), military command, judges in courts of law (Praetor Peregrinus created in 246 BC), and finally the governing of provinces. They also assumed administrative duties of consuls when these were absent from Rome. When there were more than 2 Praetors (beyond 197 BC), the additional Praetors were generally assigned as governors of Sicily, Sardinia, and the Spanish provinces (and others as province acquisition continued through the late Republic and early Principate). Like Proconsuls, Praetors could hold the title of Propraetor after their annual term of service and be appointed as provincial governors. They were entitled to 6 lictors..
Aediles (4) (from the old responsibility of caretaking of the aedes, or the Temple of Ceres):
2 as Plebeian Aediles and 2 Curule Aediles. The Plebeian Aediles were established in 494 BC along with the office of the Plebeian Tribune. Curule Aediles were originally Patrician (and a higher ranking position) and the office was established in 365 BC. Eventually the Curule Aedileship became interchangeable with Patricians and Plebes. Aediles were in charge of of such things religious festivals, public games, temples, upkeep of the city, regulation of marketplaces, the grain supply in the city of Rome while Plebeian Aediles also assisted the Plebeian Tribunes. According to the Lex Villia annalis Aediles had to be 36 years of age. Curule Aediles only were entitled to 2 lictors.
Quaestors (2-40): Quaestors typically had to be 31 years old (requirement lowered by Sulla as were all magistracies and raised back after his death) and could be Patrician or Plebeian (though in the later period this was a matter of major contention because ex-Quaestors were immediately eligible for a Senate seat). The Quaestor magistracy was developed in the time of the kings and the position in the later Republic was an evolution of various earlier positions and responsibilities. There were 2 Quaestores Parricidii, who were responsible for prosecution of criminals, and Quaestores Classici, who were financial officers and administrative assistants (civil and military). They were in charge of the state treasury at Rome and also served as quartermasters and Legionary officers under direct command of Proconsular or Praetorian Legates/Governors.
Tribunes (10) (from the Latin Tribus for Tribes): The position of the Tribune (or Tribuni Plebis) was established after the final Plebeian withdrawal from Rome in 494 BC. Naturally they were a Plebeian only position developed as a counter measure to Patrician domination in law and policy making. They were responsible for protection of lives and property of plebians; they were considered (sacrosanct) meaning their bodies were to be free of physical harm. In addition they had the power of veto over elections, laws, decrees of the senate, and the acts of all other magistrates (except a dictator) in order to protect the interest of the people (though this in itself became a powerful and manipulated political tool). They convened tribal assembly and elicited plebiscites which after 287 BC (lex Hortensia) had force of law (essentially meaning that the Tribunes could go directly to the people rather than the Senate and magistracy to propose and adopt policy).
Censors (2) (from the Latin for census): Originally established under the kings, they were elected every 5 years to conduct census, enroll new citizens, review the rolls of senate and equestrians (essentially determing eligiblilty and be sure that all criteria for inclusion were met). They were responsible for the policies governing public morals and supervised leasing of public contracts. They ranked below Praetors and above Aediles in theory and they did not have imperium or entitlement to Lictors, but in practice, this was the pinnacle of a senatorial career. It was limited to ex-consuls carried incredible prestige and dignity and was essentially the "feather in the cap" for elder statesman (at least prior to the development of various prestigious provincial governorships such as Asia Minor). Either Patricians or Plebeians (established in 351 BC) could hold the position. The office was an oddity in that the elections were every 5 years, but that they served terms of 18 months. It was the only office that had notable lengths of time without any serving magistrates and Rome often went for very long periods without a censor. It was done away with as an official magistracy in 22 BC and replaced by the title Praefectura Morum in the Imperial system.
Dictator (1): Created in 501 BC, just 9 years after the expulsion of the kings. In perilous times, typically of military emergency, public unrest or political upheaval a dictator could be appointed by originally the acting Consuls, and later by the overall senate body to have supreme authority. Typically the position was intended for Patricians, but the first Plebeian was appointed in 356 BC (C. Marcius Rutilius). The dictator appointed a Master of the Horse (Magister Equitum) originally as the name implies to lead the cavalry while the dictator commanded the legions (though the position also evolved into an administrative/executive position designed to assist the dictator). The Dictator's tenure was limited to 6 months or the duration of crisis, whichever was shorter. Generally, aside from those of Sulla and Caesar Roman dictatorships rarely lasted the entire 6 month term. Edicts of the dictator were not subject to veto and he was entitled to 24 lictors.
Lictors: Though technically not a magistrate office, the Lictors were a representation of the power of the elected magistrates over the people. Originally selected form among the plebes, they were eventually limited to freedmen, but were definitely citizens as a toga was a required uniform. The lictor's main task was to attend their assigned magistrates who held imperium: 12 lictors for consuls, 6 for Praetors abroad and 2 within Rome, dictators (24 lictors, (12 before Sulla) and curule aediles (2 lictors); the dictator's magister equitum ("Master of the Horse") was also escorted by six lictors. Men of Proconsular or Propraetorian governer rank were also entitled to lictors (the number of lictors being equal to their degree of imperium). The lictors carried rods decorated with fasces and with axes that symbolized the power to execute. They accompanied the magistrates wherever they went. If there was a crowd, the lictors opened the way and kept the magistrate safe. They also had to stand beside the magistrate whenever he addresses the crowd. Magistrates could only dispense their lictors if they were visiting a free city or addressing a higher status magistrate. Lictors also had ancient police duties: they could, at their master's command, arrest Roman citizens and punish them.