Assigning damage to multiple blockers
The latter. Each creature deals damage equal to its power to all blockers, attackers and defending players or planeswalkers, not to each of them.

Simultaneously,


The 6/6 deals 4 damage to the 4/4[1].
The 6/6 deals 2 damage to the 2/3.
The 4/4 deals 4 damage to the 6/6.
The 2/3 deals 2 damage to the 6/6.


Shortly after, when State-Based Actions are performed, the following happens simultaneously:


The 6/6 is destroyed for having damage equal to or exceeding its toughness.
The 4/4 is destroyed for having damage equal to or exceeding its toughness.





That's assuming the minimum damage is dealt to each creature. You can choose to "overkill" if you so desire, but it's rarely helpful.