The Elvis operator in Kotlin checks if the left expression is null. If it is not null, it uses that value. If it is null, it evaluates and uses the right expression instead. For example, if a variable 'name' is null, using 'name ?: "Guest"' will result in "Guest". This operator helps provide default values when dealing with nullable variables. The right side expression is only evaluated if the left side is null, which can save unnecessary computation. This behavior is shown step-by-step in the execution table and variable tracker. Understanding when the right side runs and how the operator chooses values is key to using it effectively.