Enterprise Application Integration aka EAI is all about business needs to integrate with diverse applications and systems in the enterprise. EAI includes messaging delivery, acceptance, transformation, translation, routing and business process management.
SOA is an approach to have enterprise systems built with services available and discoverable across the enterprise . ESB is an standards based infrastructure to facilate SOA implementations. It provides the messaging transformation , routing and delivery and acceptance of messages from/to various systems/back-ends using services