The idea of awareness is an interesting one (post 9) as a distinguishing factor marking the difference between the Oedipus story and the Oedipus complex.
It is of immense importance to the success of the story that Oedipus discover the identity of his mother/wife near or at the play's climax. It is this revelation which holds his doom - the revelation of how his life and identity are morally reprehensible and almost epically disastrous. If he were at all aware of "who was who", he would have been able to avert his tragedy.
For Freud there is an element of subconsciousness and therefore a lack of awareness, but the complex is rooted in an idea of fixed identities. The child is well aware of who is who.
Freud contended that a child's first love affair is with the parent of the opposite gender. There is a suspicion of others as rivals for the attentions of the parent, a possessiveness of the parent that is clearly emotional. Emotional bonding that goes beyond the parent/child relationship is also considered Oedipal and is referred to as emotional incest.
Probably everyone has covered this: the parallel between Sophocles' Oedipus Rex is that he unknowingly killed his father (Laius) and married his mother (Jocasta). So, too, Freud contended that a part of a male child's natural development was an unusual affection for his mother, while harboring resentment or jealousy for his father—probably perceived as a rival.
This concept does still remain in the mainstream of society in terms of Freud's theories, referred to in literature and cinema—if not the actual "practice," then at least the understanding of the concept in general. (Never much of a fan of Freud...)
Yes, Freud did posit that the Oedipus complex, as he coined it, is a natural stage in which young boys idolise and, in a way, "love" their mothers and therefore see themselves in competition with their fathers for the affection of their mother. Of course, the parallels between this theory and the play of Oedipus, where Oedipus unknowingly kills his father and marries his mother, made this an opportunity too good to miss for Freud.
One thing that is important to remember is that the Oedipus complex, according to Freud, was a natural stage of development (actually it was a characteristic of the phallic stage) and that the key to healthy development was successfully resolving the complex. Freud also thought that girls had an Oedipus complex too, while his student Carl Jung proposed that girls experienced an Electra complex in which they envied and feared their mothers.
One of Freud's earliest case studies (in 1909) involved a boy known as Little Hans, who the psychoanalyst determined was suffering from "castrative anxiety" and Oedipus complex, caused in part by a fear of horses, or Equinophobia. Freud came up with the term Oedipus complex after seeing a performance of the Sophocles play. Freud first coined the term in 1910.
Freud proposed the existence of an "Oedipus complex," suggesting that boys tended to see their fathers as rivals for the affection of the boys' mothers and that boys therefore sought symbolically to "kill" their fathers.