In Hinduism, maya is believed to be one of three bonds that must be cast off in order to achieve moksha (liberation of the soul from the cycle of death and rebirth) - the other two being anava (ego) and karma.
In Sikhism, maya (the world as you normally perceive it) is said to be no more manifest than a dream. Sikhism, as well as many other paths of spirituality, state that the world is like a dream, and there is nothing in it which is yours. (This last sentence has been transliterated right from the Guru Granth Sahib). An example of this is when our dreams can feel so solid and real sometimes, how do we know if we don't wake up to one every morning? What can a person actually call "MINE" in the temporary existence of a life spanning a three-quarters of a century?
A modern concept that illustrates Maya / Illusion wonderfully is the Sci-Fi Movie "The Matrix". Everything in The Matrix is believed to be real, until the character Neo wakes up, and sees that its just a dream world. One who is asleep never knows he is until he wakes up.
Maya is related to the Sunyata concept in Buddhism. Some dialogues of Plato also contain ideas reminiscent of maya, especially the famous "Parable of the Cave".