Gaudapada is the bridge figure of Advaita. Tradition makes him the teacher of Govinda, who was the teacher of Shankara, so the lineage runs Gaudapada → Govinda → Shankara without intermediary.
His one surviving work, the Mandukya Karika, is what turned the Mandukya Upanishad from one Upanishad among many into the foundational text of Advaita. In four prakaranas he develops the doctrine of ajativada, the teaching of non-origination, the claim that nothing has ever truly arisen and that what appears as world is the play of consciousness with itself.