As the other poster said, defense was part of it. The position of proconsul was actually originally a military one, created because the normal offices found themselves unable to cope with the distances involved in war with Spain. This role could be for frontier policy as well as anti-bandit operations.

However, most provinces were peaceful for most of the time. A governor of Pannonia would need to be a military man, but a governor of Greece did not. So for most governors their day to day function was as administrators, which in practical terms meant adjudicators. Law cases that local courts were unable to handle would be a major part of this, but a governor might do many other things, such as helping with building projects. Pliny the Elder as governor of Bithynia on several occasions sent for experienced engineers to help with local building projects.

In fact, the best insight you can get in to this is probably Book X of Pliny's letters. It details the correspondence between Pliny as governor and the emperor Trajan, and is very revealing regarding the day to day functions of a Roman official.

The role of the governor was to keep the security of the empire intact and keep the peace. If he was in a province where there were legions stationed then he, as the highest provincial official, was responsible for them even if there were individual legates commanding each legion. He was also the judge of last resort for the provincials. If, for instance, there was a land dispute between two cities the governor might be called in to resolve the matter among other things. Pliny the Younger, Governor of Bythinia, for instance wrote to Trajan asking him for advice what he should do about the couple of hundred Christians in his province who, while he judged them to be harmless, were being prosecuted for not offering the correct sacrifices to the emperor. Trajan wrote back saying that while Pliny should not go looking for Christians and should not accept anonymous tips if the Christians are proven to be so they should be properly punished. Another essential function of the governor was to administer the collection of taxes and ensure that they were sent off to Rome in a timely fashion.

That said, many governors did not live up to these ideals. Our histories are replete with examples of highly rapacious governors who viewed their postings abroad merely as a way to get rich quick. One example is of course Gaius Verres who Cicero prosecuted for embezzlement, but he seems like a somewhat extreme case. However, we cannot doubt that perhaps most governors embezzled taxes, extorted the natives and took substantial bribes to decide cases in favour of one party or another. The only recourse that the provincials had was a recourse to Rome, an endeavour that was quite unlikely to succeed.