I had an interesting discussion with my wife recently. We were talking about the relationship between God and Jesus - the Father and the Son. While reaching for a suitable analogy (and trying not to fall back on the clover), I was inspired to use her pregnancy.
Paul wrote: "for now we see in a mirror dimly...". We cannot and will not fully understand the Trinity entirely until we're face-to-face. We were made in His image, however, so perhaps our relationship from mother to child is an echo of the relationship of Father to Son. (Its not a perfect analogy, by far, but I hope there is credit for trying!)
A mother and her unborn child are connected - one physical body. Its not possible for my wife to leave the room without the baby or vice versa. They are, however, two separate people. Similarly, God the Father and Jesus the Son are two separate people yet share the same substance.
From here, the analogy starts to stretch, but I think its still valuable to discuss. If you understand that part, you've taken a huge step toward grasping the Trinity (as far as we can, right now). But, if I can keep rolling with the idea...
After our baby is born, he will be physically apart from my wife, yet they will still share a physical connection. His blood, his flesh, his breath all come, initially, from her body; they are connected on that genetic and cellular level even after he is born. A piece of her will always be with him. Once Jesus was born as a man on earth, He was still connected with the Father; in this case, however, it wasn't on a genetic level. It was deeper, at the spiritual level. In the same way that my son carries a part of my wife physically, Jesus carries a part of God spiritually.
Here is where we come to the big leap.
God is all-powerful, so He cannot be diluted. A little bit of God is 100% all of God. This is where the similarity ends. We are not all-powerful; a little piece of me is just a little piece. A little piece of God is GOD. Thus, when we say Jesus carried a part of God on a spiritual level, He was God. You cannot be "a little bit God" the same way my son can be a little bit my wife and a little bit me.