Edge Dashing is basically starting and ending an Initial dash at an edge, without falling off. It's mostly just to show off in a Taunt Match, but you can perform a Dash attack as well.
How To Perform
Get close to the ledge, and simply Tap as fast as you can towards the ledge. One way to do it is to hit the Control Stick to the desired direction, while letting your thumb slide off so the Control Stick returns to its original position. Your character will start a dash for one frame without falling off the edge of the platform.
Dash Attack
If you're using a Gamecube Controller, you can use the C-Stick to perform a Dash Attack next to an edge. As you hit the left Control Stick, hit the C-Stick in any direction but Up. You don't have to let go of the left Control Stick this time, you just have to make sure the C-Stick comes right after the left Control Stick.
If you try to use the A button instead of the C-Stick, you'll perform an f-smash. If using the C-Stick and you perform a d-smash, then you hit the C-Stick too soon.