In Mega Man X, the hidden Hadoken ability is called the Hadoken. It is an Easter egg inspired by Capcom's Street Fighter series and gives X a devastating energy projectile that can defeat most enemies and bosses in a single hit.
X obtains the technique from a secret Dr. Light capsule in Armored Armadillo's stage after meeting strict completion requirements. Typically, the player must defeat all eight Mavericks, collect every Heart Tank and Sub-Tank, obtain the armor upgrades, and repeatedly reach the capsule location until the special capsule appears. X must also have full health to perform the move.
Hadoken is the common English spelling used in the question, while Hadouken is another widely used transliteration and the spelling often seen in reference material. Shoryuken is the major neighboring mix-up: Mega Man X2 hides that uppercut-style technique, whereas the first Mega Man X contains the Hadoken.