The Roman Auxilia were actually an extremely diverse bunch, and despite the lack of attention were a vital part of the Roman war machine I should mention at the outset that there may be a lot of variation depending on the exact period you're interested in, so if you'd like to specific a given year then I or someone else could go into more detail.

The most well known role of Auxilia would have been in providing support troops to the heavy infantry legionaries - in particular, fulfilling roles that the Italians themselves were considered to have inadequate quality at. Most notably, Auxilia provided cavalry contingents of all distinctions, as well as missile troops like bowmen, slingers, and more. Arguably these roles are the most important ones for the Auxilia to fulfill because they're the ones the Romans themselves couldn't do well, and therefore needed specialists in.

That said, there were also Auxilia units that would be armed as infantry and who basically fight in the same manner as their citizen-soldier, legionary comrades. The quality of these men would vary from unit to unit, but some Auxilia Infantry was quite good and the Batavians in particular were considered, man for man, at least as good as the Legionaries themselves.

In the Late Empire the role of Auxilia becomes even more interesting. By this time the Imperial armies had been divided into three distinct categories - Limetanei, Comitatenses, and Palatina. I can go into further detail if you'd like, but for brevity the Palatina were considered an elite classification of soldier who were close at hand and at the personal disposal of the Emperor(s). The Palatina were a full-fledged field army that could number in the tens of thousands however, -not- an elite bodyguard unit.

In any case, serving alongside the Palatina Legions were actually Auxilia Palatina, which (depending on the unit) might have been organized as light infantry, or more heavily armed and fighting in the same style as the Legionaries themselves - again, these are Auxilia that would have been considered elite troops, and who could match the quality of the best Legionaries at Rome's disposal.

In short, Auxilia performed as virtually every sort of troop type - cavalryman, skirmisher, light infantry, heavy infantry, elite infantry and more. And, given that diversity of roles and their larger numbers they were arguably at least as integral to the defense of the empire as the legions themselves.