Plato, Platonic Idealism, and Neo-Platonism
The Platonic Legacy: What Plato Gave Western Culture
From Plato, Western thought (especially religious thought: Christian and Islamic philosophy and modern Judaism) inherits the following:
1) Idealism or The Forms: the concept of "The One" (or what Christians, Muslims, modern Jews call "God") as that eternal something which combines “the good”, “beauty” and “Truth” into a single, eternally unchanging essence, or Idea or cosmological Ideal (rather than “person” or as a deity largely served through ritual observance).
2) This Fallen World: the belief, based on the above, that everything in this corporeal, sensual world (the world we perceive thru our senses) is a "fallen" (a term Plato uses metaphorically) version of this Ideal (see "The Allegory of The Cave"). This implies that sensual knowledge – knowledge gained via the senses – can only be of this fallen world and is therefore faulty, “fallen”, and removed from the ideal. Knowledge of Truth cannot be gained from this world but only through intelligibility. In philosophy this intelligible knowledge is generally conceived of as math, or reasoning which holds to the fundamental premises of mathematical proofs. In religion, such access is through prayer and faith: ignoring this sensible/sensual knowledge of this “fallen” world and perceiving The One through the “mind’s eye” or “the soul”.
Plato thus conceives of knowledge as innate to the human psyche/soul: it arrives with the soul at birth (or conception). (This concept is fundamentally at odds, however, with nearly all modern empirically data concerning the mind and learning.)
3) Platonic (Mind-Body) Dualism: Because innate knowledge arrives before birth, the mind/psyche/soul has a separate existence from the physical body. The soul “descends” (metaphorically) and returns to from the Ideal/The One at the birth and death of body. Knowledge of the Ideal, knowledge of Truth, or "true knowledge", comes via this relationship. Plato offers the first, oldest argument that one’s physical body and soul are separate entities and that one lives on after the other has died.
4) Neo-Platonism: This refers, essentially, to the adaptation and spread of Platonic thinking throughout other cultures, philosophies and places. In the context of Western culture, it refers more specifically to this adaptation by first Hebrew and then Christian, and then Islamic, theology.
In a nutshell, in the centuries following his life, Plato (and Socrates') philosophy has a profound impact on Greek and Roman culture, and in many ways is adopted, studied accepted as truth. It then spreads throughout the Greek Empire where it has a profound effect on Jewish theology. This influence continued with the rise of the Romans and then became a centerpiece of Christian theology. Many of the central elements of Christian philosophy/theology are, in fact, explicitly originally rooted in Plato, not the Bible. See Philo Of Alexandria (20 BC - 50 AD), Plotinus (ca. AD 204–270), and to a degree, Justin Martyr (100-165).
Most importantly see Augustine of Hippo (St. Augustine) (also see here) (354-430), through whom Neo-Platonism -- and thus Plato -- is translated into Medieval Christian doctrine. Augustine On Plato, From Confessions, book VII. A good focused discussion of the relationship is here.