During the Republican Era, triumphs were generally awarded to generals who had achieved a great victory over a foreign enemy in war. Generally, a general who received this honor was a consul or a dictator. They were awarded by the senate after a proposal to award a triumph (or a request for one) was discussed and debated. I've read that the awarding of a triumph was often quite politically contentious as Roman politics was rather factional and senators were naturally resistant to awarding triumphs to rivals or members of a rival faction, however that is a generalization and I would only hope that others may have more specific examples of such machinations to back that up.

There is one example of a contentiously awarded triumph that is notable. Well into the death throes of the republic, during the dictatorship of Sulla, Pompey was awarded a triumph for a victory over King Hiarbas of Numidia in 79 BCE. Pompey was young and only an Equestrian at the time, he was also a protege of Sulla's. Sulla, always the staunch traditionalist, initially was opposed to Pompey's demands for a triumph since Pompey wasn't a consul, but relented under pressure from his supporters.

Triumphs were also often contentious because of the way they were held. Triumphs, in the republican era were like a ceremonial break from Rome's anti-monarchy values. The general would get to be dressed up as if he were king for a day and enjoy some semblance of splendor and honors that aren't quite fitting with the legacy of Cincinnatus. To make it tolerable, the general was expected to remember that the "glory was to Rome" and to act with due humility. On top of that the occasion wasn't supposed to be too over-the-top. A variety of triumphs pushed the envelope on all counts through the Republican Era, with Pompey's triumphs being particularly offensive to those who worried about Rome's backsliding from republican traditions of modesty, humility, and austerity.