The Oedipus complex is a child's desire to want the opposite-sex parent and subsequent desire to compete with the same-sex parent for the opposite-sex parent.
Mum and Dad
Regardless of how many identifications a person can take on, the most powerful and long lasting will be the ones developed during childhood between the Mum and Dad.
The Oedipus Complex
This is what the Oedipus complex looks like:
A boy loves his Mum, because she feeds him and cares for him. At the same time, the boy looks up to his Dad and wants to be just like them, because they've won the attraction of Mum. But over time, as the boy wants the Mum more and more, they start to see the Dad as an obstacle getting in the way, and so they start to develop a sense of hatred towards their Dad.
This works vice-versa for a girl. The girl loves her Dad and develops a hatred towards her Mum.
How the Ego deals with the Oedipus complex
Eventually, the child's Ego understands that it's inappropriate to love their parents in such an intense way, and so the Ego starts a process of identification to extinguish this intense love. As a result the child then starts to identify with either their Mum or their Dad.
Usually, boys identify with their Dad to give them a confident sense of masculinity and girls identify with their Mum to give them a confident sense of femininity.
However, it's also common for girls to start identifying with their Dads once they have to give up their Dads as a love object. The same is vice-versa with boys and their Mums.
The spectrum
It's important to point out that the Oedipus complex isn't so rigid. Children naturally have both masculine and feminine tendencies, so it's common for a boy to both love his Mum and envy his Dad, whilst also loving his Dad and envying his Mum at the same time.
Over time, the Oedipus complex fades away and people tend to fall somewhere along on the following spectrum:
At one end, we've got the normal complex where the child likes the opposite-sex and rivals the same-sex, at the negative end, we have the opposite where the child likes the same-sex parent and hates the opposite sex.
This effectively leaves behind four strands (hate for mum, love for mum, hate for dad, love for dad) that leads to two identifications:
Identity 1
At the same time:
Identity 2
So after this they get two identifications, whichever one is strongest depends on whether the child has more masculine or feminine tendencies.
Either way, these two identifications form the core of the ego, called the ego-ideal or super-ego. It acts as an inner judge for other contents in the ego.