An alternate character (or “EX character”, etc.) is a character that is a different version of the same character, chosen through some deliberate means.
The practice begins with Real Bout Fatal Fury Special for SNK, and with Street Fighter Alpha 2 for Capcom.
These characters are referred to with some indicator prepended to their name, such as “EX”, “Ura” (after the Japanese), or “Alt.” (which I prefer).
After the early examples, there came alternate characters that actually have something to do with the story, such as Orochi Iori.
Also, some series have alternate characters for practically every character, such as the Rasetsu mode from Samurai Shodown III & IV, the Street Fighter Alpha 3 “ism” system, the Capcom vs. SNK “groove” system, and so forth.
Many alternate characters are darker versions of the base version; they tend to have latent powers awakened, and act maybe slightly evil. As far as the gameplay goes, the way their moves work and which ones are available are basically different, and often mirror older versions of that character’s move set.
Many alternate characters appear as secret characters, being fitted with more damaging moves, and being better aggressively, but on the other hand (as with many high-throughput characters), their defense is low, and are easier to make dizzy.