[SalesForce] how to create a custom button for a custom object on a related list

This question is very similar to the one you already asked here: how to create a custom button for a custom object on a related list

If I don't misunderstand you requirements, you got a lot of things conceptually wrong.

First Competitors are related to quotes and you want to find all competitors for that quote. So you have to bypass the QuoteId as filter criteria - not the CompetitorId.

So your button code could be something like

/apex/CompitetorsForQuote?id={!Quote.Id}

There is also absolutely no need for using Javascript as Button type. You should use URL.

Then you query for a single competitor (using the wrong where) but you should query for a list.

string QuoteId = System.currentPageReference().getParameters().get('Id');
Competitor__c[] comps = [SELECT Id,Name,Product_Series__c,   Part_Number__c,  
    Price_Offered__c , Volume__c,  Date_Price_is_valid__c FROM Compitetor__c 
    WHERE Quote__c = :QuoteId;
];

This is probably not all and my code is untested and may contain errors. But it could help you to think about the structure.

If you mean that you want to get the selected records in the controller then you can use the methods of the standard controller such as getSelected like this:

public  class CompitetorsEditExt {

    private ApexPages.StandardSetController sc;

    public CompitetorsEditExt(ApexPages.StandardSetController sc) {
        this.sc = sc;
        sc.setPageSize(10);
    }

    private Compitetor__c[] getSelected() {
        return (Compitetor__c[]) sc.getSelected();
    }

    private void someOtherMethod() {
        Compitetor__c[] competitors = getSelected();
        ...
    }
}