I think you don't mean the domain name, the domain name doesn't appear there, it's the site name.

try this:

define('PAGE_TITLE_FORMAT', '%1$s :: %2$s');

i added the code using html to the header and it did not work. where should i put it

that is your site's name which is called by the template.

you can change the text from dashboard >sitewide settings

if you dont want it to display at all, you need to find the header.php file in your theme and remove it there (editing source files)

I cannot try the change until I can access it.

now i need to know what to put where

I have found this and think this is my site header .php in my themes. How do I edit it so that I remove just the text for the site name in the header nav bar? Thank you.

<?php
defined('C5_EXECUTE') or die(_("Access Denied."));
$this->inc('elements/header.php'); ?>
<div id="page">
<div id="headerSpacer"></div>
<div id="header">
<h1 id="logo"><!--
--><a href="<?php echo DIR_REL?>/"><?php
$block = Block::getByName('My_Site_Name');
if( $block && $block->bID ) $block->display();
else echo SITE;
?></a><!--
--></h1>
<?php
// we use the "is edit mode" check because, in edit mode, the bottom of the area overlaps the item below it, because
// we're using absolute positioning. So in edit mode we add a bit of space so everything looks nice.
if (!$c->isEditMode()) { ?>
<div class="spacer"></div>
<?php } ?>
<div id="header-area">
<?php
$ah = new Area('Header Nav');
$ah->display($c);
?>
</div>
</div>
<div id="homeHeader">
<?php
$ahh = new Area('Header');
$ahh->display($c);
?>
</div>
<div id="central">
<div id="sidebar">
<?php
$as = new Area('Sidebar');
$as->display($c);
?>
</div>
<div id="body">
<?php
$a = new Area('Main');
$a->display($c);
?>
</div>
<div class="spacer">&nbsp;</div>
</div>
<div id="footer">
<span class="powered-by"><?php echo t('Built with concrete5')?> <a href="http://www.concrete5.org">CMS</a>.</span>
&copy; <?php echo date('Y')?> <a href="<?php echo DIR_REL?>/"><?php echo SITE?></a>.
&nbsp;&nbsp;
<?php echo t('All rights reserved.')?>
<?php
$u = new User();
if ($u->isRegistered()) { ?>
<?php
if (Config::get("ENABLE_USER_PROFILES")) {
$userName = '<a href="' . $this->url('/profile') . '">' . $u->getUserName() . '</a>';
} else {
$userName = $u->getUserName();
}
?>
<span class="sign-in"><?php echo t('Currently logged in as <b>%s</b>.', $userName)?> <a href="<?php echo $this->url('/login', 'logout')?>"><?php echo t('Sign Out')?></a></span>
<?php } else { ?>
<span class="sign-in"><a href="<?php echo $this->url('/login')?>"><?php echo t('Sign In to Edit this Site')?></a></span>
<?php } ?>
</div>
</div>
<?php $this->inc('elements/footer.php'); ?>