To save existing records, use EDIT mode in the force:recordData component. The controller markup is as follows:

<force:recordData aura:id="recordComponent"
      recordId="{!v.recordId}"
      layoutType="FULL"
      targetRecord="{!v.record}"
      targetFields="{!v.simpleRecord}"
      targetError="{!v.recordError}"
      recordUpdated="{!c.handleRecordUpdated}"
      mode = "EDIT"
      />

The controller function is as follows:

 component.find("recordComponent").saveRecord($A.getCallback(function(saveResult) {
            // NOTE: If you want a specific behavior(an action or UI behavior) when this action is successful 
            // then handle that in a callback (generic logic when record is changed should be handled in recordUpdated event handler)
            if (saveResult.state === "SUCCESS" || saveResult.state === "DRAFT") {
                // handle component related logic in event handler
            } else if (saveResult.state === "INCOMPLETE") {
                console.log("User is offline, device doesn't support drafts.");
            } else if (saveResult.state === "ERROR") {
                console.log('Problem saving record, error: ' + JSON.stringify(saveResult.error));
            } else {
                console.log('Unknown problem, state: ' + saveResult.state + ', error: ' + JSON.stringify(saveResult.error));
            }
        }));

It is important to use aura:id to find the target component. The syntax to find the component with the aura:id value recordComponent is as follows:

component.find("recordComponent")

The force:recordData component consists of a function, saveRecord, that saves the data and provides a callback function to handle SUCCESS and failures. The following lines of code save the record, using saveRecord: 

component.find("recordComponent").saveRecord($A.getCallback(function(saveResult) {
     
}