[SalesForce] Is Process builder a complete alternate for the triggers

From what it seems you are dealing with a recursion problem here. The most obvious solution I can think of is using some custom settings or using a helper class that would enable the execution of the trigger one time and that would become inactive after the execution.

This would guarantee that your trigger is only being executed once.

This is an example of a helper class

public Class checkRecursive{
    private static boolean run = true;
    public static boolean runOnce(){
    if(run){
     run=false;
     return true;
    }else{
        return run;
    }
    }
}

And this is in action

trigger updateTrigger on anyObject(after update) {

    if(checkRecursive.runOnce())
    {
    //write your code here            
    }

}

The above example was taken from here : link

In the process builder, have one more criteria to the node of the process builder, such as update description only if it is null or if the field has changed. Having these kind of checks prevents multiple updates in one transaction.