From the store you can choose to start reading the book immediately in your web browser with Kindle Cloud Reader, or you can choose to open the book with your Kindle for iPad app.
In case you were wondering, it doesn't work with Android browsers (unless you first change the user agent). I checked.
BTW, did you know that you can add a link to the iPad Kindle Store to your iPad's desktop? Click on the icon in the upper left corner that is the 4th from the left. You'll see the option in the menu that drops down.
So first Amazon released the Cloud Reader, and now the iPad has its own Kindle Store. yep, it really does look like Amazon is trying to get around Apple. That would be good, except that Amazon is almost as much of a control freak. And to be honest, Amazon did this for their own benefit, not yours.