How did people from different civilizations/countries communicate when they spoke completely different languages?

Well not necessarily Columbus, but Cortez was able to communicate with the Native Americans despite in a very difficult way.

After a Spanish priest, Geronimo de Aguilar, accompanied by his crew shipwrecked on Yucatan and were captured by some Mayan soldiers in 1511. His crew was sacrificed to the Mayan gods while others died from diseases. But de Aguilar and another Spaniard named Gonzalo Guerrero were able to escape. They were later caught by another Mayan tribe and taken as slaves. Both de Aguilar and Guerrero learned the Mayan language during their eight years of captivity.

When Cortez landed in Mexico he learned about these Spanish prisoners. He decided to track them down, and de Aguilar decided to join Cortez in his journey through Mexico. Guerrero on the other hand decided to stay with the Mayans, he later even married with a Mayan woman and had a child.

So, de Aguilar was able to speak both Spanish and Mayan. But the language of the Aztecs was Nahuatl. Luckily enough, Cortez found a Nahua woman who was held as a slave when he landed in Mexico. This woman was named Malinalli, but the Spanish called her La Malinche. She was able to speak both Mayan and Nahuatl. When Cortez arrived in the Aztec Empire, he was able to communicate with Montezuma. Montezuma would speak in Nahuatl, La Malinche would translate this to Mayan and de Aguilar would translate it from Mayan to Spanish.

Of course this is just a small example, but during the Lewis & Clark expedition, a same kind of way was used to communicate with the natives. If you were abducted by aliens, you would now their language after a certain amount of time, since you are kinda forced to learn it. People who are taken prisoner during wartime are often known to have learned the language of their captors.

Source: Gordon, Helen. Voice of the Vanquished: The Story of the Slave Marina and Hernan Cortes.