Indian rubber refers to a rule present in many childrens' games which are both played with balls and require players to be tagged out. Think kickball, baseball (though usually not when played with an actual baseball), etc.

Indian rubber is getting someone out by hitting them with the ball, or with whatever thrown implement is being played with.

On the playground there is usually an argument between the rougher kids and the gentler kids as to whether or not the game being played will feature indian rubber. This argument is usually settled either by force of numbers, or physical threats.