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Your answer definitely falls within the realm of AJAX. If you are using a form, try setting the onsubmit eventListener equal to return false; (plus any other needed attributes) like so:

<form onsubmit="return false;">

Then you send a post and parse it with PHP and depending on your desired effect, and the input of the user, you echo out a response to your waiting AJAX script, which can take the ajax.repsonseText and depending on whether or not the response was equal to what you were looking for, you can set the desired HTML element’s innerHTML equal to you whatever you want to without refreshing the page. More than likely something like this:

if(ajax.responseText != "error") {
    document.getElementById("userInputDiv").innerHTML = ajax.responseText;
}