In The First Man in Rome, there are squabbles about who should become a pontifex, as well as pontifex maximus, after either post has become vacant, and families are jealously trying to preserve their prerogatives.
The pontifex maximus was the highest priestly position in Rome, head of the college of the pontifices.
POʹNTIFEX (ἱεροδιδάσκαλος, ἱερονόμος, ἱεροφύλαξ, ἱεροφάντης). The origin of this word is explained in various ways. Q. Scaevola, who was himself pontifex maximus, derived it from posse and facere, and Varro from pons, because the pontiffs, he says, had built the Pons Sublicius, and afterwards frequently restored it, that it might be possible to perform sacrifices on each side of the Tiber (Varro, de Ling. Lat. v.83, ed. Müller; Dionys. ii.73). This statement is, however, contradicted by the tradition which ascribes the building of the Pons Sublicius to Ancus Martius (Liv. i.33), at a time when the pontiffs had long existed and borne this name. Göttling (Gesch. d. Röm. Staatsv. p173) thinks that pontifex is only another form for pompifex, which would characterise the pontiffs only as the managers and conductors of public processions and solemnities. But it seems far more probable that the word is formed from pons and facere (in the signification of the Greek ῥέζειν, to perform a sacrifice), and that consequently it signifies the priests who offered sacrifices upon the bridge. The ancient sacrifice to which the name thus alludes, is that of the Argeans on the sacred or Sublician bridge, which is described by Dionysius (i.38; compare Argei). Greek writers, moreover, sometimes translate the word pontiffs by γεφυροποιοί . . .
The pontifex maximus was the president of the college and acted in its name, whence he alone is frequently mentioned in cases in which he must be considered only as the organ of the college. He was generally chosen from among the most distinguished persons, and such as had held a curule magistracy, or were already members of the college (Liv. xxxv.5, xl.42). Two of his especial duties were to appoint (capere) the Vestal virgins and the flamines [Vestales; Flamen], and to be present at every marriage by confarreatio. When festive games were vowed or a dedication made, the chief pontiff had to repeat over the persons who made the vow or the dedication, the formula with which it was to be performed (praeire verba, Liv. v.40, ix.46, iv.27). During the period of the republic, when the people exercised sovereign power in every respect, we find that if the pontiff on constitutional or religious grounds refused to perform this solemnity, he might be compelled by the people. Read all
The college originally was composed solely of patricians, but the lex ogulnia in 300 BCE abolished that prerogative, and plebeians were added. The above article goes in detail into the subject of election versus cooptatio of these priesthoods, a rather confusing sequence of events.
The last elected pontifex maximus was Caius Julius Caesar. He violated the law that said that the pontifex maximus may not leave Italy, but he was by no means the first one to do so. Plutarch – explicitly and eloquently – reports (Tib. Grac. 21.3) that the pontifex maximus Publius Nasica left Italy, permanently, in 133 BCE.
Beginning with Augustus, after the death of Marcus Aemilius Lepidus,
the position automatically went to the emperor, and more or less turned
it into the status of 'high priest'. Eventually, in 381 CE, the
emperor Gratian declined to use the title of pontifex maximus, and it was accepted by the pope Damasus I (wiki, handle with care) and has remained one of the titles of the Bishop of Rome ever since.
Wikipedia also has an entry on Pontifex Maximus.
Until Augustus, the pontifex maximus always resided in the domus publica, adjacent to the Atrium Vestae, residence of the Vestal Virgins (Samuel Ball Platner's Topographical Dictionary of Ancient Rome, with images), who were under his oversight.