Plato's main study in philosophy was his theory of forms also known as his theory of ideas. Although there is the material world in which we live, Plato also believed that there was an "eternal world" of concepts and forms. He believed that this world was more real than the material world and it is the object of knowledge, not opinion.
One of the things that we have to distinguish in Plato's ideas is the distinction between the Forms and the material world. The Forms are the ideal or "perfect equivalent" of everything that exists. The material world is the world in which we live which consists of everything being reasonable approximations of an ideal Form. Plato believed that for every single thing there is in the material world, there is a perfect form of it in an ideal world or "realm." One of Plato's examples of this is his idea of beauty. He explains that true beauty cannot exist in the material world because the material world is full of opinion as people can say "her nose is too big" or "his ears are too small" when others might think differently. Plato goes on to explain that true beauty can only be a matter of opinion as everyone has a unique idea of beauty.
Plato gave reasons for suggesting that there are Forms. One of these reasons is that it suggests that we go to this ideal realm when we die where everything is perfect or "ideal." This is very closely related to Christianity in the sense of there being and eternal and perfect afterlife which Christians obviously relate to as being heaven. Plato also suggested that when we describe particular objects with certain words we are relating to the world of Forms. For example, if one were to say, "They are truly brilliant." They would clearly be relating to the world of Forms as we do not know what true brilliance is as we have never witnessed or experienced it.
Plato used a number of ways to illustrate his ideas of the Forms. One of his particularly interesting illustrations was when he related perfection to a circle.