Running Elasticsearch on AWS involves a detailed process of provisioning and configuring EC2 instances to create a three-node Elasticsearch cluster using version 2.3.3. The guide outlines steps such as selecting the appropriate Amazon Machine Image (AMI), configuring instance details, setting up storage, tagging instances, and configuring security groups. Users are advised to install the necessary plugins, including the AWS cloud plugin, and to configure Elasticsearch to optimize performance, particularly by adjusting the JVM heap size. Starting up and verifying the cluster involves ensuring all nodes have the same version and configuration, with the aim of achieving a "green" status for optimal operation. While managing Elasticsearch on AWS requires a hands-on approach, Elastic Cloud offers an alternative with a managed service that simplifies deployment and management, supporting features like security, automatic scaling, and backup, and is available for a free trial without a credit card requirement.