Chain Combos are the repeated use of a weak kick/jab and using a stronger attack or special move immediately after. With the combo-friendly atmosphere of Street Fighter Alpha, characters have been given quick jabs and weak kicks that can be canceled into itself or another move quickly, in most cases right after the foe gets hurt.
Most characters in Street Fighter Alpha only have one move that can initiate a Chain Combo, due to the speed it requires. Perhaps the most famous Chain Combo is Ryu/Ken's weak crouching kick used repeatedly, often followed with a roundhouse or Hadoken; when done correctly it is inescapable. Other characters usually rely on their jabbing punch for Chain Combos.
The concept of chain combos was almost completely removed in the game's sequel, Street Fighter Alpha 2. Only Guy and Gen have a sort of Chain Combo. The rest of the cast had their weak/jab moves made slower, allowing for fewer, weaker combos to balance out certain characters. However, "A-ism" in Street Fighter Alpha 3 allows Chain Combos.