form with no action and where enter does not reload page

You'll want to include action="javascript:void(0);" to your form to prevent page reloads and maintain HTML standard.

Add an onsubmit handler to the form (either via plain js or jquery $().submit(fn)), and return false unless your specific conditions are met.

Unless you don't want the form to submit, ever - in which case, why not just leave out the 'action' attribute on the form element?

Simply add this event to your text field. It will prevent a submission on pressing Enter, and you're free to add a submit button or call form.submit() as required:

onKeyPress="if (event.which == 13) return false;"

For example:

<input id="txt" type="text" onKeyPress="if (event.which == 13) return false;"></input>

When you press enter in a form the natural behaviour of form is to being submited, to stop this behaviour which is not natural, you have to prevent it from submiting( default behaviour), with jquery:

$("#yourFormId").on("submit",function(event){event.preventDefault()})

an idea:

<form method="POST" action="javascript:void(0);" onSubmit="CheckPassword()">
    <input id="pwset" type="text" size="20" name='pwuser'><br><br>
    <button type="button" onclick="CheckPassword()">Next</button>
</form>

and

<script type="text/javascript">
    $("#pwset").focus();
    function CheckPassword()
    {
        inputtxt = $("#pwset").val();
        //and now your code
        $("#div1").load("next.php #div2");
        return false;
    }
</script>