The Roman army changed through the centuries. So it is impossible to say anything about it that would apply to all times and places. What I mention below is all from the Imperial period, from Augustus to around the fourth century.

The Romans had Auxilia which where formed of allied non-Roman soldiers. These made up most of the specialized units in the roman army including skirmishers, scouts, missile troops, and cavalry. And some of them could get pretty unique based on where they were raised (slingers vs archers or camel cavalry vs horse). But they wouldn't be considered elite by any means.

Each legion had a double sized cohort, which was also made of the best most experienced and capable centuries, and commanded by the senior commander. This gave each legion a unit of unusual strength and toughness. So in a pinch every legion had an "elite" unit organically attached to it.

In addition to the regular legions there were four specialized forces. The first you have already mentioned, the Praetorian Guard. This developed from the bodyguard a legionary commander would have. Augustus was well aware that the tradition of not stationing legions in Rome itself left it vulnerable. Sulla, Marius, Caesar and others had all taken advantage of this fact. So he created the Praetorians to be a guard for himself and the city. Out of respect for the tradition, they didn't normally wear armor inside the city and kept their weapons hidden. They were supposedly an "elite" unit. And their performance suggests they were better than the average legion, at least at times. But they were still just a legion.

The second specialized unit was the Urban Cohorts. These were created to protect certain places and people (like the treasury) and to act as a heavy police unit. Prior to their creation it wasn't uncommon for street gangs to rule the city. They were still considered to be "legionaries." But they rarely saw actual battle.

The third was the Vigilum. They were the light police force and the fire brigade for Rome. They saw combat even less than the Urban Cohorts.

The fourth unit, which only lasted less than a century, was the Germani custodes corporis. These were a Germanic imperial bodyguard. And were trusted in that role precisely because they weren't Roman. But they were disbanded after only a century. Once it became more common to gain the throne via revolt, the new emperors didn't like keeping bodyguards around from the former regimes. Galba disbanded them in 68. But the idea was so useful most emperors ended up with a troop of German Cavalry for their personal guard.