The item at the right, encountered in a Kentucky woods, is the Giant Puffball, Calvatia gigantea. The species gets even larger than this one, but finding one just this large is quite a thrill -- especially because this species is great to eat! Just slice and fry.
True puffballs produce no stalk -- they look like round or pear-shaped balls simply lying on the ground. Their outer walls, which often are so weak that you can poke your fingers through them, contain the puffballs' contents, which consists of a homogeneous mass of spores. This mass, which often but not always is white, is referred to as the spore mass, or gleba. At first the gleba is soft, dry and fleshy but eventually it grows tougher and becomes powdery. The powder is made up of spores.
When a puffball matures, its spores escape either through the body's ruptured walls, or through a hole, or pore, at the top, referred to as the ostiole. The old puffball at the left, probably the genus Lycoperdon, clearly shows the ostiole at its top. The yellow-framed inset in the image's lower left side shows a cross-section of the puffball.
In general, puffballs, which are quite common, are good to eat. However, in North America the genus Scleroderma might possibly make some people sick. It forms a purple gleba from the first, so it's easy to identify. Just don't eat any puffball whose gleba is not white, soft and fresh-looking.
Also, some poisonous stalked mushrooms, such as the deadly Amanitas, emerge from a button stage which might look like a puffball. However, if you cut such a button stage down the middle, you'll see the typical mushroom shape inside and know that it's not a puffball, and therefore you should be careful about eating it.
The picture at the right shows a special kind of puffball, the Earthstar, genus Geastrum. This was found along the forest trail leading to my little home in the woods. At first the Earthstar looks a lot like the Lycoperdon, but then its outside wall splits and opens up, its "arms" curl back as shown, and this reveals a regular puffball inside. This interior puffball then develops spores inside it, and those spores escape through a hole in the top just as in regular puffballs.