1) Define a region in your theme. This region will allow blocks to be added to it. Surround the region in an div that has an ID which you can specifically target to make the whole region floating.
For example, in your .info file, you may define it like this:
regions[floating_region] = Floating region
And in your page.tpl.php, you would add the region like this:
<?php print $floating_region; ?>
As you can see, the floating region is wrapped in a div with an ID of #floating_region
2) Make the region float in your CSS. You do this by setting position fixed to the region, and defining where you want the region to be placed:
bottom:0; // this fixes it to the bottom of the screen
width:100%; This will set the region to be the full width of the screen
Now anything you add to the region will float at the bottom. You will probably need to adjust the CSS and add a height and maybe some other stuff, but this is the main process by how you will define it.
Also wouldn't the fixed position make it so that it is below the footer not at the bottom of the browser?
When you set the position to fixed, it takes it out of the flow of the document, and fixes it in a fixed position of the screen, regardless of scrolling. This means that the footer position doesn't matter whatsoever. I believe the default for all browsers when you set position fixed is the top left of the screen, so if you don't set 'bottom' and 'left' values, that is where it will be fixed to, regardless of where it appears in your HTML.