ball
The word 'palla' in Italian is used in much the same way as 'ball' in English. It could refer to a ball used in sports or to any spherical or near spherical object. It can also be used figuratively, for instance in idioms. Please note that in Italian all nouns have a gender, 'palla' is feminine, hence you use 'la' (the) before it.
Let's go play ball.
This sentence is using 'palla' in a context of playing a ball game. Different kinds of games can be played with a ball, hence the exact game is not specified.
I lost my ball in the park.
In this example, 'palla' is used to indicate possession of a ball. The speaker is expressing that they lost their ball in the park.
The ball is in the opponent's court.
In this context, 'palla' is being used in the context of competitive sports. It's a figurative phrase that means the opponent now has the responsibility or control.