Is it possible to hide or not display the post-kicker on a particular page?
The page I want to hide the post-kicker on has an id of 425.

I've tried this, but it doesn't work:

body.page page-id-425 #post-kicker {
display: none;
}

body.page page-id-425 #post-kicker

That syntax is incorrect. what it says is for any element with an ID of post–kicker that is a child of an element called page–id–425 in the body of the page with a class called page apply the CSS. If you wanted to use both classes “page” and “page–id–425” you would have to say body.page.page–id–425 but since both page and page-id-425 art classes you can just say body.page–id–425

Thanks for explaining that.
It looks like I should be able to use the below, but it isn't working.
Am I missing something? I also tried it without !important

body.page–id–425 #post-kicker {
display: none !important;
}

Well is post-kicker a 'ID' or an 'class'? If it is an ID you use the # but if it is a class you use a period

post-kicker is a class.
Maybe I've still got the syntax incorrect:

body.page–id–425.post-kicker {
display: none !important;
}

yup you still have it incorrect. the format is
element#idname or
element.classname

so you can use something like 'element#idname element.classname' which can be shortened to
element#idname .classname or
#idname element.classname or even
#idname .classname

when you say 'body.page–id–425.post-kicker' you are saying "apply this to a body that has a class of page-id-425 and a class of post-kicker" you need a space in there

I've tried that, too. No good.

body.page–id–425 post-kicker {
display: none !important;
}

http://dev.advantagecardservices.com/application-form/
Application Form, How Can We Help, the background image behind these, shouldn't be there.

body.page–id–425 post-kicker {
background-image: none !important;
}

Didn't work. I'd like to not show the entire post-kicker. Any other ideas?

I don't see the post kicker showing