It's common in military parlance to give a short, unambiguous designation to your enemies. Typically the NATO Phonetic Alphabet is used for this type of designation. In situations where your squad may draw from many nationalities, dialects, or ethnicities with different pronunciations of common words, this reduces the chances that a message will be misinterpreted.
For instance, "tango" is used by the military in many situations in place of "enemy target." Likewise, squads may be given names based on the first letters of the alphabet, such as Alpha/Alfa, Bravo, Charlie, Delta, Echo, and Foxtrot.
The "X" in X-COM stands for "Extraterrestrial" so it follows that when you're engaging the enemy, you're fighting "X"es, or "X-Ray" in the phonetic alphabet.