Hi Kirtan, here's a better version of the trigger code. What I've done differently, is establish variables (lines 5 - 10 in the code below) that set variables to store the points assigned to each Role. The reason I've done this is because if the points will change in the future, you can easily go back into the code and adjust the points right there in those variables at the beginning, instead of trying to find the score assignments throughout the code.

I've also added the the scoring logic within the Account's Contacts loop (lines 40 - 61 in the code below).

As a final result you have a trigger that looks like this:

trigger accountTotalScore on Contact (before insert, before update) {

    Set<Id> accIds = new Set<Id>();
    List<Account> accToUpdateList = new List<Account>();
    // set the points for each role
    Integer surgeonScore    = 10;
    Integer fireAssistScore = 10;
    Integer csrScore        = 20;
    Integer nurseScore      = 5;
    Integer adminScore      = 0;

    // add accountIds to a set
    for (Contact myCon : Trigger.new) {
        if (myCon.AccountId != null) {
            accIds.add(myCon.AccountId);
        }
    }

    // get all accounts related to Contacts entering the trigger
    if (!accIds.isEmpty()) {
        List<Account> myAccounts = [
            SELECT Id, Total_Score__c, (SELECT Id, Role__c FROM Contacts WHERE Role__c != null)
              FROM Account
             WHERE Id IN :accIds
        ];

        if (!myAccounts.isEmpty()) { 
            // Loop through each Account
            for (Account myAccount : myAccounts) {
                // initialize variables
                Boolean surgeon    = false;
                Boolean fireAssist = false;
                Boolean csr        = false;
                Boolean admin      = false;
                Boolean nurse      = false;
                Integer score      = 0;

                if (!myAccount.contacts.isEmpty()) {
                    // Search for the Role values in each Contact
                    for (Contact accContact : myAccount.contacts) {
                        if (accContact.Role__c.containsIgnoreCase('Surgeon') && !surgeon) {
                            surgeon = true;
                            score  += surgeonScore;
                        }
                        if (accContact.Role__c.containsIgnoreCase('Fire Assist') && !fireAssist) {
                            fireAssist = true;
                            score += fireAssistScore;
                        }
                        if (accContact.Role__c.containsIgnoreCase('CSR') && !csr) {
                            csr = true;
                            score += csrScore;
                        }
                        if (accContact.Role__c.containsIgnoreCase('Nurse') && !nurse) {
                            nurse = true;
                            score += nurseScore;
                        }
                        if (accContact.Role__c.containsIgnoreCase('Admin') && !admin) {
                            admin = true;
                            score += adminScore;
                        }
                    }
                }
                myAccount.Total_Score__c = score;
                accToUpdateList.add(myAccount);
            }
            if (!accToUpdateList.isEmpty()) {
                update accToUpdateList;
            }
        }
    }
}