Does anyone know how to "Burn" the edges of a picture or blur out the edges of a picture using Elements 6?

The word you're looking for is Vignette - though it's not even listed in the PSE Help files!

Two main methods.......

With PSE6 you have, under Filters, "Correct Camera Distortion" - perhaps that's not a helpful name for all that you'll find in there, but if you go there you'll find, near the top, "Vignette", with 'Darken' and 'Lighten'. Give those a tweak and you'll find the dark version, where the image is enclosed a little by shadows, and the light version, where the image fades to the edges.

The other way is to add a fill layer on top and then "knock out" the middle. Make an adjustment layer, choose "Solid Color" and choose black (or any other colour for that matter but it's best in either black or white). With the rectangular marquee selection tool, draw a rectangle well inside the edges of the image (like 15% in from each edge). Go to Select > Feather and give it lots of feather pixels (around 250 for a full, straight-from-camera image but you may need to try other settings). Fancy trick now..... Hold Alt down and hit Backspace - Voila! Now you'll discover if you've got enough Feathering - in most cases, you won't, so you'll have to go back to that step and reset the number.

Oh no, I know how to do a vignette, but this is a burning effect kind of like "bokeh" or the blurred effect you get in pictures that is normally obtained by having alot of depth of field in a picture.

I'm pretty sure, I did this as one of my homeworks so you might want to do a search. Others may have added different options. Lot''s of different ways to do the same thing. This is how I do mine.

New blank layer.
Edit, fill, choose 50% gray from the drop down box.
Change your blend mode to hard light.
Choose the burn tool at midtones around 16%.
Burn the edges,

Oh, OK. Or did you mean the artificial representation of shallow depth of field?
If so, you can duplicate the background and then use Gaussian Blur on that layer, then use a Layer Mask (whichever way you do that) and erase some of the blurring at the main subject or point of focus. That one seems to often be called the "Tilt/Shift" effect, or "Villiage".

Thanks- I'll give it a shot.