You only need a simple image editor such as MS Paint, I use MS Paint.
For an alternative, see this Instructable
Step 1: The Shrink
Use the rectangular Select tool to select the area you wish to pixellate.
Look to the lower right of the Paint window, the dimensions of the area will be shown as you drag the mouse. Select an area that is a whole multiple of 10 (see step 3 for more on this).
From this point onwards don't touch the image or the right hand toolbar with your cursor - only use the menus.
From the Image menu, choose Stretch/Skew... (CTRL-W)
Reduce the size of the area both horizontally and vertically. Here I am making the area 10 x smaller by selecting 10%.
In doing this shrink there will be 100 times fewer pixels in the selection.
Step 2: The Stretch
Paint can't stretch selections more than 500% (5 x) so to increase the selection back to it's original size I need to stretch by 500%, then repeat the stretch at 200%.
If the finished result isn't what you wanted, note what isn't right about it, hit CTRL-Z or From the Edit menu choose Undo and start again.
Step 3: Fine Tuning
If the pixels are too large, you have shrunk the selection too far. If you think they should be half the size, only shrink half as much.
If the pixels are too small and they don't obscure the image enough, the selection needs shrinking more. As above if you think the pixels should be twice as big, shrink the selection twice as much.
This is more subtle:
You'll notice that in this series of images the guy on the right loses an eye. The original selection was 71 x 72 pixels reduced to 8 x 8 pixels - Paint has rounded 7.1 x 7.2 pixels up. When this was stretched back the selection ended up as 80 x 80, a little over the original size.
This is why the dimensions of the selected area are important. Look to the lower right of the Paint window as in step 1, and select 'round' dimensions which will scale back easily.
All this being said, the effect still worked.