[jQuery] Post to PHP file, update div

[jQuery] Post to PHP file, update div


I'm trying to figure out something that I know how to do in regular
javascript however I've had a heck of a time figuring out how to do in
jQuery.
<div id="updateme"></div>
<form id="form1" onsubmit="return false;">
<input type="text" id="value1">
<input type="submit" id="submit" value="Submit" />
</form>
What I want to do is to pass value1 to a PHP file that processes it,
then updates the div "updateme" using the data from the PHP file
without refreshing the page. It's such a simple concept but I have yet
to figure out how this is done.
Could someone post the jQuery code to do this?