Here is the sample page and controller. You can change the object names to fit your requirement.

Visualforca page:

<apex:page controller="SelectCompetitorsController" sidebar="false" showHeader="false">
<apex:includeScript value="https://ajax.googleapis.com/ajax/libs/jquery/1.7.2/jquery.min.js" />
<script type="text/javascript">
    $ = jQuery.noConflict();
    function SelectAccounts(chk, accountId)
        {
            var accountIds = $('input[id$="accountIds"]');
            var selectedAccounts = accountIds.val();
            if (chk.checked == true)
            {
                if (selectedAccounts == '')
                {
                    selectedAccounts = accountId;
                }
                else
                {
                    selectedAccounts = selectedAccounts + ',' + accountId;
                }
            }
            else 
            {
                var accountIndex = selectedAccounts.indexOf(accountId);
                var accountToRemove = '';

                if (selectedAccounts.length == accountId.length) {
                    accountToRemove = accountId;
                }
                else {
                    if (accountIndex == 0){
                        accountToRemove = accountId+ ',';
                    }
                    else{
                        accountToRemove = ',' + accountId;
                    }
                }
            
                selectedAccounts = selectedAccounts.replace(accountToRemove, '');
            }
            
        accountIds.val(selectedAccounts);
       // alert (accountIds.val());
    }
</script>
    <apex:form >
        <apex:inputHidden value="{!selectedAccountIds}" id="accountIds"/>
        <apex:pageBlock title="Select Competitors">
            <apex:pageBlockTable value="{!accounts}" var="acc">
                <apex:column headerValue="Select" width="100px;">
                    <apex:inputCheckbox onclick="javascript:SelectAccounts(this, '{!acc.Id}');"/>
                </apex:column>
                <apex:column headerValue="Account Name" value="{!acc.Name}"/>
            </apex:pageBlockTable>
            <apex:commandButton value="Add Competitors" action="{!addCompetitors}"/>
        </apex:pageBlock>
    </apex:form>
</apex:page>

Controller:
public class SelectCompetitorsController {

    public List<Account> accounts {get; set;}
    Map<Id, Account> accountMap {get; set;}
    public string selectedAccountIds {get; set;}
    string parentAccountId;
    public SelectCompetitorsController()
    {
        parentAccountId = ApexPages.currentPage().getParameters().get('accId');
        accountMap = new Map<Id, Account>([select Id, Name from Account]);
        accounts = accountmap.Values();
    }
    
    public PageReference addCompetitors() {
        List<string> accountIds = selectedAccountIds.split(',');
        List<Competitor__c> competitors = new List<Competitor__c>();
        
        for (string s : accountIds)
        {
            Competitor__c competitor = new Competitor__c(Account__c = parentAccountId, Name = accountMap.get(s).Name);
            competitors.add(competitor);
        }
        if (competitors.size() > 0)
        {
            insert competitors;
        }
        string strPageRef = '"javascript:window.close();"';
        return new PageReference(strPageRef);
    }
}

You need add a list button on your competitor object and in the Behaviour section select "Execute JavaScript" and use the following code in text area below.

window.mypopup = window.open('/apex/SelectCompetitors?accId={!Account.Id}','mypopup','top=150,left=200,location=1,status=1,scrollbars=1, width=500,height=600'); 
window.onfocus = function() { 
if (window.mypopup.closed) { 
window.location.reload(); 
} 
};

Hope the above solution works for you.