Chain Combos are performed through the repeated use of a light kick or punch, followed by a stronger attack (including unique attacks and special attacks) or special move immediately after. (e.g. jab into fierce, jab into roundhouse, etc.) Most characters in Street Fighter Alpha have one move that can initiate a Chain Combo, due to the speed it requires.
The Chain Combo system was almost completely removed in the game's sequel, Street Fighter Alpha 2, as most of the cast's light attacks were made slower in order to balance out the game; only Guy and Gen retain the ability to perform a Chain Combo throughout the series, replaced as a Target Combo within the series.
Chain Combos, now known as Alpha Combos in Street Fighter Alpha 3, is a selectable and unlockable feature in its console versions.
Chain Combos are predated by the Darkstalkers series, where it was first featured. Later game series' such as the crossover series made extensive use of this feature, though the timing is eased significantly, and is a staple game mechanic.
Tying into the Alpha Combo example, a majority of other fighting games have their own spin on the Chain Combo/magic series mechanic, often allowing for basic blockstrings during pressure, as well as each game series following their own set rules for what attacks can chain-into-what when it comes to the character and their normal attacks in question (such as which normal can chain into which other normal or a Unique Attack then into a special). These games may or may not have their own unique names for their Chain Combos in that regard.
Street Fighter X Tekken uses a Cross Rush mechanic that is performed similar to a Chain Combo, with regular attacks used in increasing strength order to create a combo that automatically launches a foe; if it lands successfully, the user will also tag their partner in to continue the combo.