Dell Inspiron 570 - you have one of these click here? It's a regular-sized PC case (i.e. not slim-type) right?
What kind of space are we looking at where the graphics card (GPU) needs to be fitted? See this pic to know what I mean: click here. The horizontal measurement is where the GPU is likely to be slotted in, so measure directly over the PCI-E (GPU) slot).
Do you have installed, or will you be installing any expansion cards below the graphics card? If yes, how many slots available. And how many exp cards are, or will be, installed? This is important too as your new GPU's girth is likely to occupy 1-2 additional exp slots below it.
Unlike gengiscant ATI suggestions, I'd opt for a nVidia-based GPU - a GTX 460 (click here) perhaps, as it's a little more future proof in terms of features (Review click here). Like he says, I reckon a PSU replacement is a must. MY suggestion would be for something like this Be Quiet 530W click here. But consider a more powerful one if you think you'll be upgrading the GPU at some point to a GTX 480 / HD 5970 for example. Ask for suggestions if required, as PSUs come in various qualities, sizes & prices. G