Interpreters and mediators were a requirement for diplomacy, trade, travel, and more throughout history just as they are now.

Particularly for the big empire builders like Alexander and Genghis, it simply wasn't practical to learn the countless languages and dialects within their realm. Large numbers of officials would have existed to interact with these groups. Often, they'd be people learning the language of the conqueror rather than the other way around.

I'm glad you brought up Cortes since the conquest of Mexico and its interpreters are really well documented.

There are two interpreters that matter for the Spanish conquest of Mexico. Geronimo de Aguilar was a Spanish priest that learned Maya during captivity on the Yucatan Peninsula. When Cortes came in 1519, Aguilar joined him as a translator. (1) The more important one was La Malinche. She was a Yucatecan Mayan woman that joined Cortes but spoke only Maya and the Aztec's own Nahuatl language. She's a fascinating subject on her own, if a controversial one, and might well be the most famous translator in history. Initially then, Cortes spoke through Aguilar who translated it for La Malinche who then passed it on to the Mexican leaders. This would have been how the famous meeting between Cortes and Montezuma transpired. (2) Eventually La Malinche learned Spanish and Aguilar was unnecessary.

We know quite a lot about La Malinche, relatively speaking. She's mentioned in the writings of the conquistadors, where she's credited with foiling Mexican plots against the Spanish. (2) She's even more prominent in the fragmentary native oral histories that were recorded in the first few years after the conquest, (3) which makes sense since the Mexica would have been understanding and hearing her so much. It also makes us rethink how these encounters would actually have taken place. The official accounts of the meeting between Cortes and Montezuma is one of flowery extended speeches which almost certainly didn't happen, mainly because we know they were working through multiple translators that simply wouldn't have been able to do that. (4)

There's far too many translators to count in history - virtually all diplomatic interaction would have had them. Translators often doubled as informants too, and vice versa. Information was key to any empire, whether building one or maintaining one, so people that knew the area and knew the language were invaluable.

In terms of how people engaged with those whose language they didn't share, you get a wide spectrum of approaches. There were some rulers that did make an effort to learn the languages of their people. Other times it simply would have been too difficult to try.

One interesting approach was that taken by the Romans. Especially in the Republican period (note that Rome certainly had an empire long before that period ended) Roman leaders themselves were never supposed to speak the language of 'lesser' peoples. For example, Cato the Elder visited Athens during the Roman-Seleucid War but, even though he spoke Greek, gave a speech only in Latin and let the translator repeat it to the audience. (5) There's plenty of strange episodes like this in Roman history. The best might be the meeting of Scipio Africanus and Hannibal on the day before the Battle of Zama in 202 BC. Both of them could speak Greek and could easily have spoken directly but tradition meant Scipio couldn't use any language but Latin in his official role, so both of them used translators from their native tongues. (6)

On the subject of missionaries, this opens a wider debate over the language of religion. Often, religious texts come in one language. The Bible was in Latin for Catholics, the Qu'ran in Arabic for most Muslims. Even in the cases where the texts were translated, many of the learned and devout would have preferred it in the original tongue. That being said, in terms of conversion practicality always won out over principle.

The Catholic missionaries in the Americas faced a major issue over how to convert. Some like the Franciscan Geronimo de Mendieta believed that teaching the natives Spanish was essential to any conversion, not to mention 'civilising' them from barbaric native tongues. (7) For obvious reasons however, this wasn't practical. The few hundred missionaries learning the native languages was much more realistic that the millions of natives being taught Spanishor Latin. Obviously that took time to do, but within a few decades of conquest you see conversion efforts clearly done in native languages. Take Peru for example where Domingo de Santo Tomas produced a Quechua grammar and learning guide specifically for the purpose of aiding missionaries. (8) Tides did shift against native languages over time, but only after Spanish colonial society had grown - and indigenous populations had shrunk - enough for Spanish to be the dominant language already. (9)

One last thing on rulers making themselves understood in new lands. Empire and conquest weren't nearly as much about wiping away the old as people often think. Often, the very top layers of rule were eliminated while keeping as much of what was left as possible. If a region you're conquering already has a strong infrastructure, effective tax system, thriving trade, and stable social order, why change it? For example, The Mongols didn't create a new Mongol world order; they pretty much just imposed a nominal supremacy over everyone with some taxes and let people go about their business like before. That usually meant preserving existing practices and the languages they were done in. The impetus was on the newly conquered administrators and officials to learn the language of the new rulers in hopes of gaining favour and ingratiating themselves. Very few empires ever attempted a linguistic unity project, it's simply not feasible and often not necessary. At most, the centre's language becomes a lingua franca that people choose to pick up, as is the case with Latin, or colonisation by the empire's own people coupled with extinction of natives leaves a language as dominant, as is the case with many of the European languages in the Americas and Africa.

Language is remarkably powerful. It's so easy to overlook but being able to use the language in some form was often make or break for conquest or travel. Past societies depended on translators to conduct all sorts of things, just like we do today.

References

(1) H. Cortes, 'First Letter', in A. Pagden (ed. and trans.), Letters from Mexico, (Yale, 1986)

(2) Bernal Diaz del Castillo, The True History of the Conquest of New Spain Vol 2, (Liechtenstein, 1967), p12

(3) Miguel Leon-Portilla (ed.), Broken Spears: The Aztec Account of the Conquest of Mexico, (Beacon Press, 2006)

(4) F. J. Brooks, 'Motecuzoma Xocoyotl, Hernan Cortes, and Bernal Diaz del Castillo: The Construction of an Arrest’, The Hispanic American Review Vol 75 No 2, (1995), p165

(5) Plutarch, Lives, Cato the Elder, 12

(6) Polybius, The Histories, 15.6.3

(7) J. L. Phelon, The Millennial Kingdom of the Franciscans in the New World, (U. of California P., 1970), p87

(8) S. MacCormack, Religion in the Andes: Vision and Imagination in Early Colonial Peru, (Princeton U. P, 1991), p447-8

(9) Religion in the Andes: Vision and Imagination in Early Colonial Peru, p450-1