What, for Plato, are the forms? Nothing could be more difficult than to answer this question: even Augustine, who certainly knew Plato much much better than I, dumurs in attempting to “nail” Plato “down:” after all, “he emulated … the well known practice of his master Socrates … of concealing his own knowledge of opinion.” (City of God VIII.4)
So, then, what follows is the best I can do in stammering toward some kind of sketchy impression of Plato’s doctrine of the forms.
Many interpreters of Plato, including Aristotle, who followed the truth over his friend, take Platonic forms to be mental concepts. And, one can see a certain plausibility in this interpretation, especially based on the early dialogues, such as the Euthyphro, where the form of “piousness” is considered in the context of a discussion on genus and species, and in the Phaedo, where equal sticks and stones are said to be equal, due to the reality of some form of equality. In the former case it can be said that genus and species are, in fact, mental concepts, and in the latter case one can say that mathematical type entities seem to be mental concepts, as well.
So, is this the answer? For Plato, are the forms simply mental concepts?
Not quite, for beginning in the middle dialogues, including the Republic, the Symposium, and the Phaedrus, the Good seems to be a mysterious reality, in which the human soul participates. This is construed in diverse ways, but in each case the Good is transcendent and elusive while nonetheless characterized as a kind of desiridatum which the human minds strives for. In the Republic one thinks of the Good which lies above the divided line, and hence above the rational intellect simply speaking. Since the entire logic of the divided line proceeds on the basis of participation (with the lower realm of appearances “participating” in the upper realm of rationality), it makes sense to see rational nous, or indeed the whole soul or the whole person, as participating, striving to participate, in the good, into which it is “wooed” or even seduced.
Speaking of seduction, in the Symposium we encounter divine Diotima, for whom not only does the lover enjoy or delight in looking upon the beloved. In addition to this, Diotima (according to Seth Bernardete in his commentary on the Symposium) completes Socrates’s life-long education, providing the third of his major breakthroughs or epiphanies (after the second sailing of the Phaedo and the Parmenidean realization, narrated in the first half of the Parmenides, that if the forms are knowable, they are knowable only by God), that all things participate in Beauty. On this scenario, the ultimate form is “Beauty” (there is room in Plato to argue that beauty is “convertible” with the Good), and all things participate in it, as illuminated reason is enabled to perceive.
Hence, a more plausible view, in opposition to form-as-mental-concept, is the view that Plato regards the Form(s) as participata, that in which all things—including the human mind—participate. For Plato, then, it is in the Form(s) that we live and move and have our being.
If ever there were an “extra-mental” reality, this is it.
 The Greek eidos is cognate with the verb for “to look,” also with the Latin video, whose first letter was originally a diagamma, yielding the Greek stem “id.”