I think the question in the title and the question in the body are quite different. As far as playground is concerned, both in or on are fine.
In the ground, as Sirish points out, has the literal meaning of being in the ground, as in:
There are lots of minerals in the ground.
He found a beautiful stone in the ground.
On the ground has two meanings. One is the literal meaning of being on the surface of the ground, as in:
There's a big rock on the ground.
On the ground has also an idiomatic meaning: among the general public, as in
Their political ideas have a lot of support on the ground.
