<p>I have a winforms application, that why someone clicks on a button I need to open up IE to a specific URL.</p>

<p>When someone closes the winforms app, I then need to close IE.</p>

<p>Is this possible? If yes, how?</p>

It would be possible to do that, but it might be a better idea to simply embed Internet Explorer into your application using the WebBrowser control. That way when you close the website you have no chance of closing the whole window when your website opened in a new tab in an existing IE window.
Edit:
If you're going to do it anyway,  look at the
MSDN page on System.Diagnostics.Process.Close