The Pennsylvania Dutch called it Cats and Mice and the midwest and southern states called it Scrap Bag.
I agree. I love this block! So much so that I have a pattern called Fluttering Around based on the Beggar's Block that I designed in early 2011 as one of the designs for my Double Nickel Quilt Challenge.
There is a whole post, from July 2012, about the Beggar's Block at Quilt Inspiration . There is also an old pattern here.
I made my 6" finished Beggar's Block differently than the larger block used in Fluttering Around. I paper pieced the smaller block like this and it was very easy! The center is all one piece!
You can download the pattern for the 6" finished block paper pieced block here. The Beggar's Block can also be found in Electric Quilt's Blockbase program as #1758b.