Alex,

You have several issues here. Primarily, since the trigger is on Commission__c and you're wanting to update the affected Contact records, you need to issue a DML statement on the Contact records that are changing (aka. update(contacts)).

However, you're solution isn't very scaleable -- You're processing ALL accounts and ALL their contacts, even if the commission didn't change.  You should probably get a list of accounts (or contacts) from the Commission object and only query those affected records to begin with.

When changing the primary contact, you should really ensure that you're making a change and NOT issue the update if it's the same contact that was already primary.

NOTE: Untested code. Use/modify at your discression and own the risk:

This example assumes there is an account lookup field on the commission__c object called Account__c. 

I suggest using a map for other records to update like Contact records in this scenario.  This will ensure that you can only list a single Contact once in the list of records to update. It's probably not needed in this example because each contact should only be evaluated as the primary contact 1 time, but it's a handy way to prevent issues.

trigger primaryContact on Commision__c (after insert, after update) {
     
    Set<Id> accountIdSet = new Set<Id>();
    for(Commision__c com : Trigger.New){
        if(com.Account__c!=null) accountIdSet.add(com.Account__c);
    }

    List<Account> accToCon = [SELECT Id,
                              (SELECT Id, Name, Total_Commission__c FROM Contacts) FROM Account
        WHERE Id in :accountIdSet];

   Map<Id,Contact> updateContacts = new Map<Id,Contact>();

    // do not need to loop through commissions again, because we have all affected accounts now
        for(Account c: accToCon){
            Contact oldPrimaryContact;
            Contact newPrimaryContact;
            if(!c.Contacts.isEmpty()){
                oldPrimaryContact = findPrimaryContact(c.Contacts);
                newPrimaryContact = findMaxCommission(c.Contacts);
                System.debug(oldPrimaryContact);
                System.debug(newPrimaryContact);
                // only update contacts if they need to change
                if (oldPrimaryContact!=null && newPrimaryContact!=null && oldPrimaryContact.Id!=newPrimaryContact.Id) {
                    updateContacts.put(oldPrimaryContact.Id oldPrimaryContact);
                }
                if (newPrimaryContact!=null && (oldPrimaryContact==null || oldPrimaryContact.Id!=newPrimaryContact.Id) {
                    updateContacts.put(newPrimaryContact.Id, newPrimaryContact); 
                }
            }
            else{
                continue;
            }
            // need to perform DML to update the contacts
            update(updateContacts.values());
        }
     
    Contact findMaxCommission(List<Contact> contacts){
        Contact chosenContact = contacts[0];
        Decimal max = contacts[0].Total_Commission__c;
        for (Contact i : contacts) {
            if (max < i.Total_Commission__c) {
                max = i.Total_Commission__c;
                chosenContact = i;
            }
        }
        return chosenContact;
    }

    Contact findPrimaryContact(List<Contact> contacts){
        Contact chosenContact;
        for (Contact i : contacts) {
            if (i.Primary__c) {
                chosenContact = i;
            }
        }
        return chosenContact;
    }

    }

}