If you want to learn how to use the link thisto get the element of the parent element to pass it to removeChild, then it is quite simple, just use parentNode:
<div class="divbutton">
<button type="button" style="display: none;" id="i" onclick="document.body.removeChild(this.parentNode)">Hello</button>
</div>
However, since you are using jQuery, it makes sense to use it:
$(document).on('mouseenter', '.divbutton', function () {
$(this).find(":button").show();
}).on('mouseleave', '.divbutton', function () {
$(this).find(":button").hide();
}).on('click', ':button', function() {
$(this).parent().remove();
});
#divbuttonX .divbutton, CSS .
: http://jsfiddle.net/5qfjo0c7/