1) first check if guest checkout is disabled:
I think the solution to your problem is that you haven't allowed the guest checkout from your magento backend. To do this, go to
system> configuration> checkout and set "Allow Guest Checkout" to "Yes"
2) if step 1 does not work:
http://stackoverflow.com/questions/24566168/magento-shipping-information-the-continue-button-not-working
go to /app/design/frontend/base/default/template/checkout/onepage/payment.phtml
in line 36:
change from :
<fieldset>
<?php echo $this->getChildHtml('methods') ?>
</fieldset>
to:
<fieldset id="checkout-payment-method-load">
<?php echo $this->getChildHtml('methods') ?>
</fieldset>