First, some terms:

praenomen = "first name" (Marcus, Gaius, Titus, Sextus, Quintus, etc)

nomen = family name, the middle one (Iulius, Cornelius, etc)

cognomen = the third one, used to identify branches (Caesar, Sulla, etc)

A great reference data set for this question is the large dossier of Cicero's letters, in which we get a wide variety of uses of names of individuals, some with tria nomina and some without a cognomen. In general terms, all three names are only used together by Cicero in specific contexts, though there are quite a few exceptions to these. First, in letters of recommendation, the tria nomina is pretty common. Second, he tends to use all three when there might be confusion, especially about historical persons who suffered from a particularly confusing situation of homonyms. Third, he uses all three in contexts in which he wants to highlight the illustrious ancestry of the referent (or the opposite).

Other than that, the tria nomina is relatively rare, but not completely absent. For the rest, he uses a combination of two of the three elements. There are generally four possibilities:

praenomen + nomen (Marcus Licinius);

praenomen + cognomen (Marcus Crassus);

nomen + cognomen (Licinius Crassus);

and the odd (for us moderns) cognomen + nomen (Crassus Licinius).

For the small circle of Roman aristocrats who had very recognizable nomina and cognomina, the nomen was usually dropped. So "Gaius Caesar" or "Publius Sulla" or "Marcus Scipio" or "Quintus Cicero." The nomen was obvious to anyone at the time, and Latin has a tendency to reduce for efficiency whenever expedient. There were some exceptions, when the nomen stubbornly sticks around (ex: P. Sulpicius Rufus, never "Publius Rufus," perhaps to avoid some possible confusion with the Caelius Rufus clan, or the like). Using the praenomen + nomen was also incredibly unhelpful in many cases. In the very few instances in which Cicero used it, we moderns often have a very hard time figuring out who is meant (like the very vague "G. Iulius," which could be half a dozen people). In other words: the cognomen was useful for differentiating, and the cognomen automatically indicated the nomen in most cases.

Only aristocrats had the tria nomina in most cases. The vast majority of Roman citizens just had two names, or even just a nomen. Some had a pseudo-cognomen earned or coined in their lifetime, and therefore not functionally the same as a cognomen like Rufus, which was many generations old. For these, Cicero oscillates between using both names, or just one, as the case may be, or inverting the two.

For family and close acquaintances, especially those shared by sender and receiver of the letter, any and every combination is possible, without concern for ambiguity. When Sally writes to her mother about her brother Mark, she need not clarify. Mark is her brother, by default, in that context. It was the same in most cases for the Romans, as well. When Cicero was addressing the court in his defense of Marcus Caelius Rufus, any combination was clear.

edit: if anyone really wants to punish themselves and dig deep into this, the venerable D. R. Shackleton Bailey produced an onomasticon for both the letters of Cicero, and also separately for his speeches, both of which have extensive introductory remarks and notes. Therein you can find wonderful Roman names, like Aulus Aurius Melinus; Quintus Fufius Calenus; Publius Nigidius Figulus; and Lucius Plaetorius Cestianus.