The Theory of Forms (also known as the Theory of Ideas) was the centrepiece of Plato’s philosophy. It is essentially the belief that everything on Earth is an inferior copy of an original, supreme and heavenly master-copy. In effect, it amounts to a philosophical counterpart of the popular religious concept of the fallen paradise.
It is important to recall that Plato sees a form as the ideal essence of something, a transcendent entity that is perfect, immutable, indivisible. The things of our everyday world are imperfect copies of the forms; they are multiple, but the forms themselves are one. For example, there are many different kinds of cats: some have black fur, some grey, others orange. There is, however, something that all cats have, namely, cat-ness. According to Plato, the many cats are merely a facsimile of the form Cat.
Socrates expounds the theory of ideas [forms]; he is sure that there are ideas of likeness, justice, beauty, and goodness; he is not sure that there is an idea of man; and he rejects with indignation the suggestion that there could be ideas of such things as hair and mud and dirt -- though, he adds, there are times when he thinks that there is nothing without an idea.
Plato introduces the Forms by distinguishing between what he calls "fanciers" and true philosophers. [...] The fanciers, he says, can appreciate beauty when it is found in physical manifestations of beauty, but they cannot appreciate Beauty itself. These fanciers, Plato says, are dreaming. On the other hand, those who can see Beauty itself, understand the essence of Beauty are philosophers, and are awake.
Plato believed that there exists an immaterial Universe of `forms', perfect aspects of everyday things such as a table, bird, and ideas/emotions, joy, action, etc. The objects and ideas in our material world are `shadows' of the forms. This solves the problem of how objects in the material world are all distinct (no two tables are exactly the same) yet they all have `tableness' in common. There are different objects reflecting the `tableness' from the Universe of Forms.
...many have understood the recollection of Forms as an act that involves summoning innate ideas of the Forms up from the recesses of the mind that perceived them before birth. In order to retrieve a Form from the intelligible realm of immutable ‘being’, it is generally believed that the mind must shun and see past those mutable instances of the ultimate essences that clutter the sensible realm of ‘becoming’. Only in this way can the human intellect conceive a thought that corresponds to the way things really are.
[Plato's dialogue, Parmenides] represents Socrates as claiming to have originated the theory of the derivation of the physical world from a plurality of noon-physical Forms which Plato had already ascribed to him in the Phaedo. In both dialogues the conception of individual universal terms as names of substantial Forms is related explicitly to the ontological controversy of the mid-fifth century.
The concept of organizational form lies at the center of theories of organi- zational ecology and evolution. Yet most theorists have treated the status of the concept of organizational forms as unproblematic. As is often the case with core concepts, analysts have been comfortable with an intuitive understanding.
Plato makes a distinction between the sensible world and the supersensible world; he calls the former the “visible” and the latter the “intelligible.” The visible is further divided into shadows or images and their corresponding objects; for example, the shadow, reflection, or painting of a tree and the tree itself. He divides the intelligible into mathematical objects and the Forms; for example, conceptual ideas of circles are located in the former, and the ideal, universal circle is located in the latter.
The status of appearances now came into question. What is the form really and how is that related to substance?
Thus, the theory of matter and form (today's hylomorphism) was born. Starting with at least Plato and possibly germinal in some of the presocratics the forms were considered as being "in" something else, which Plato called nature (physis).
...the Theory of Forms establishes that there is a single, eternal and indivisible Form corresponding to every property. This implies that things are many as well as divisible at least in the sensible world, and each of them participate of as many Forms as the properties that constitute the sensible thing.