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Stated briefly, Plato's Theory Forms is that for every abstract idea there is a corresponding Form. For instance: there is a Form of Beauty, a Form of Justice, a Form of Courage, and so on. (Arguably, this isn't just true for abstract ideas, but of material objects as well. So, for instance, there is a Form of apples, a Form of chairs, et cetera.) Anything in the sensible world that has a particular quality only has that quality by virtue of the fact that it participates in the Form that is associated with it. While the sensible world is imperfect and perishable, the world of Forms is a transcendent reality of perfection and immortality. The goal of the philosopher, according to Plato's theory, is to properly understand and appreciate these Forms, freeing the philsopher's soul from the mundane material world.
However, Plato is never very clear on what precisely he means when setting forth his Theory of Forms in Phaedo. There are three main interpretations of how something participates in a Form (as suggested in the Analysis discussion of the Summary & Analysis for section of 100b - 102d). First, there is the view that Forms are paradigms. Under this interpretation, the Form of Beauty is the paradigm of beauty, and a beautiful person participates in this form by imperfectly resembling the paradigm of beauty.
An alternative reading is the view that Forms are universals. Under this interpretation, the Form of Beauty is the quality that all beautiful things share in common, that which makes them such that we can say that they are beautiful. A beautiful person would then participate in the Form of Beauty by possessing that shared quality that makes things beautiful.
Lastly, there is the view that Forms are stuffs. Under this interpretation, the Form of Beauty is the sum total of all instances of beauty in the world, and a person would participate in the Form of Beauty by possessing some of the "stuff" that is beauty.
