Add class to DIV depending on page loaded

Question

You might want to consider using the WordPress function body_class()

<body <?php body_class($class); ?>>

Then you could use the body class as your current page reference.

This function is for example used by the Twenty-Twelve theme.

For a single post it will become

<body class="single single-post ...">

and for the home page:

<body class="home blog ...">

You can read more about this function in the Codex here:

Edit:

I know you can do this with for example body_class() if you want to
add classes to BODY. But how do I do this with a DIV?

You could use it in your div tag:

<div <?php body_class(); ?>></div>

but it should be sufficient to have this only in the body tag.