I remember in one of my classes, my professor struggled to explain the concept of imperium. He told us that imperium was the authority to command men/legions, but that it also had a higher, religious significance as well. How is imperium as an idea interpreted? What did it mean to the Roman people? Did it change or develop over time? Thanks.
Imperium is a very complicated concept; it is unlike anything we know these days.
Origins
At it's core, imperium was the power of the Roman kings of old. Imperium was the power of the King to act in the public interest. As the authority of the King, imperium was indivisible. One person had the power to do whatever was necessary, with only minimal safeguards against abuse of power.
A very closely related concept were the auspicia publica, "public auspices" or just auspices. This is the theological aspect your professor was talking about. It is quite literally bird watching, with the birds giving a sign from the gods. According to legend Romulus performed the first auspiciation on the day he founded Rome. Twelve vultures sent by Jupiter confirmed Romulus had the correct location for the city.
Imperium and auspices are closely related because for every action to be undertaken the omens must be favourable. If imperium is the power to act on behalf of the people, the auspices is the power to act in accordance with the will of the gods. One cannot go without the other, or disaster will surely follow.
Early Republic
After the expulsion of the kings, the Romans devised a system which prevented too much power from concentrating in any one person. In this early system, every year the Senate would elect two consuls. Each consul would have imperium and full auspices. At this time, this basically meant that the consuls had the power of the kings - power checked only by the fact that they could veto each other and by the fact that they were appointed for only a year. Additionaly safeguards were added over time though. The People's Tribunes got the authority to veto the decisions of any magistrate (including consuls) and in times of crisis, if the consuls failed to protect Rome (or died trying), a dictator could be appointed. The dictator would have the highest imperium and auspices for six months, replacing the two consuls.
Perhaps you have noticed that that last sentence has something interesting. I say "highest imperium", not just imperium. This is because over time, the concept of imperium shifted. When the power went from the King to the consuls, imperium was given to two people. Both had full imperium, but you could say that imperium was split. It was no longer something indivisible possessed by one person, but it was something which could be spread around, so to speak. If this is done once (when the position of consul was created), it could be done more often, and so it was. It is not very practical to give a lot of people complete power though, so we see several processes taking place regarding imperium.
Spreading imperium around
The first is that a second grade of imperium is invented. The two consuls had consular grade imperium. As Rome grew though, the consuls could no longer take care of everything. The position of praetor was established to take over some tasks of the consuls. Praetors would be elected yearly just as consuls were. At first there was one praetor, whose task was the administering of justice. Having a praetor take care of justice freed up the consuls to do other things (mainly command armies overseas). The praetor's task required imperium, as it originally was something the king did. But as I said above, it would be impractical to just give everyone the same imperium, the same power. So the Romans invented a lower grade of imperium for the praetors. It was imperium and it came with auspices, but it wasn't the imperium of the consuls. Consuls could overrule praetors at any time.
The difference between the consular imperium and the imperium minus of the praetors is not very big. Essentially the only difference is that the imperium minus is defined to be lesser than the consular imperium. But praetors could use their imperium for the same stuff consuls did. The first praetor had the task of presiding over civil cases, but he had the authority to command armies if necessary. Over time more praetors were elected every year, six in the middle Republic. Two praetors for justice (for Romans - the praetor urbanus - and for foreigners - the praetor inter peregrinos) and several praetors to govern provinces (Sicily and Sardinia at first, two provinces in Spain from 197 BC).
Extending imperium.
The spreading of (lesser) imperium to more people wasn't enough to govern the quickly expanding empire though. The Romans came up with another idea to have sufficient people to take care of everything which had to be done, without resorting to electing more and more people every year. From the third century onwards, the Senate could decide that an outgoing magistrate could continue to hold imperium for another year. The Senate would still elect two new consuls, but one or both of the previous consuls would be allowed to operate as proconsul ("acting in place of a consul"). This meant that he could continue whatever he was doing (usually fighting a war) while the new consuls would receive other tasks. Such an extension might happen several years in a row. Later on the same would apply to praetors, allowing some to stay on as propraetor.
The interesting thing here is that this extended imperium would only apply for a specific task. That is, whereas a consul had unchecked imperium any place outside Rome (with a very strong suggestion what to do with it), a proconsul could only finish what he was doing - usually fighting a war. No shenanigans would be accepted here. Finish the job and return to Rome to lay down the imperium. What you see here is that over time, imperium came to be something not just limited in time (given year by year) but also limited in geographic scope. A proconsul or (pro-)praetor has imperium to govern the province assigned to him. That's it.
(This all about the normal development of imperium. In the Late Republic proconsulships and dictatorships start getting abused by powerful individuals. This is a whole different cup of tea, for another wall of text post.)
A final note on the distinction between imperium and auspices
Imperium is definitely the most interesting concept of the two, as it has changed so much over time. The auspices haven't changed a whole lot over time. The Romans were very conservative where these kind of things were concerned. That's why I haven't said a lot about it above. I do want to note an interesting tidbit though.
As you know the consuls and praetors were elected. This was done by the Comitia Centuriata or Centuriate Assembly. It was via these elections that the consuls and praetors received their imperium. As we have seen anyone with imperium would also need auspices. The auspices however were not given by the Centuriate Assembly. It was another assembly, the Comitia Curiata or Curiate Assembly which had to separately grant auspices to anyone with imperium. This effectively meant that the Curiate Assembly had to approve the elections by the Centuriate Assembly, as without auspices an elected person could do nothing.
I hope all of this makes any sense the way I explained it. :) If you have any further questions, do let me know.
Sources:
I have used this earlier answer of mine (under my previous name) as a basis for this post, though I've added/rewritten so much it's hardly recognisable.