The Alpha Djinn is the original Djinn from which all other Djinns come.
In his bid to find the location of Purgatory, Crowley sent members of the Campbell Family and Sam Winchester on a hunt for the Alphas. They captured lesser monsters in order to lure their respective Alpha out.
Samuel manages to capture Brigitta, a djinn. She is taken to Crowley's prison where Crowley has her tortured extensively in hopes that she will tell Crowley where her "daddy" is, referring to the Alpha.
Crowley never gets his answer since he is distracted by the Winchester brothers. Brigitta is presumably killed later on. The Alpha Djinn's fate is unknown.