The Data Guard Monitor (DMON) process in Oracle Data Guard is responsible for performing several critical tasks. It performs role transitions when switchover requests are made, communicates with DMON processes in other database instances that are part of the broker configuration, and communicates with the DMON process of the observer to monitor a primary database in case a fast start failover is required. These tasks help ensure the correct functioning of the Data Guard environment by managing role transitions, ensuring smooth communication between different instances, and actively monitoring for failover requirements.