Leela means an act, play.
Example 1 - The cosmic sport is divine Leela. The struggle of creatures across the cycle of life and death, the veil of ignorance, realization and transcendence, all this forms a part of that Leela. It is called Leela in the sense that it passes over the subtle body of beings without making a difference to the actual state of existence of either the self or Brahman.
Example 2 - God incarnate is called "leela manusha vigraha" or having assumed human body as a Leela.
Example 3 - Dream is said to be remembered in a leela fashion, meaning that it does not make a difference to the actual state of existence but a passes over the mind.
Leela were actions of Bhagawan, Saints, Sants, and great masters that were used to teach their disciples different principles and subjects. The objective of many of their miracles was/is to convey the teaching in a language that their disciple could best understand. However, due to the fact that people differ in their capacity and ability to learn and interpret the happenings around them, the same leela will often have different imports to different disciples and observers. Each disciple draws his own conclusion and learns a lesson for himself from one common context or word of the sadguru.