Technically, any Catholic male who has reached the age of reason, is not a heretic, is not in schism, and is not notorious for simony can be elected pope there is no other requirement for election (although there are several requirements before a person can actually assume the papacy once elected). It might even be technically possible for them to elect a non-Catholic male, if they had reason to believe that he would immediately convert to Catholicism.
The lack of a long list of formal requirements is probably due to the fact that, in times past, it was possible for the elector cardinals to elect a new pope not through formal ballots but rather through sudden acclamation after being inspired. A list of formal rules would make such acclamation much more difficult, even though the rules have now eliminated acclamation (as well as the use of committees) to elect new popes.
In practice, of course, Catholic laity and even common clergy have no real chance to be elected pope and the papacy is restricted to cardinals or perhaps a few bishops. The last non-cardinal elected pope was Urban VI in 1379. Certain cardinals may be more likely to be elected than others (because of age, for example), but within that group there is no way to say who is the favorite.
Indeed, it may be more likely that a non-favorite could be elected. Every favorite may be favored by a different group but no group may be able to get the others to accept their candidate. As a consequence, the man finally elected may be no ones favorite, but ultimately the only man that enough of the cardinals can actually agree upon.
In another informal nod to tradition, the next pope will certainly have to speak Italian. Most people regard the pope as simply the head of the Roman Catholic Church, and that he is; but we must not forget that he is also the Bishop of Rome, and as such he carries with him the same responsibilities of all bishops. Indeed, no one can become pope officially until they are also officially made bishop in Rome.
One of the sources of the the great popularity of Pope John XXIII was apparently the fact that he acted like the Bishop of Rome more than most popes. He visited prisons, visited hospitals, and took a genuine interest in the lives and fortunes of the average Roman citizen. This was as unusual as it was appropriate and it helped guarantee his place in the hearts and minds of Romans for generations to come.
If the next pope cannot address the crowds in Rome in their own language, he wont be readily accepted or highly regarded. This may not be the mob of antiquity, but it seems unlikely that the elector cardinals will completely ignore their needs when it comes to choosing the next pope. The exclusion of non-Italian speakers may not narrow the field of likely popes very far, but it does narrow it.