"Wherefore henceforth know we no man after the flesh: yea, though we have known Christ after the flesh, yet now henceforth know we him no more" (2 Cor.5:16).
Paul is saying that after he was converted he no longer evaluated people according to their outward appearance, and telling those in the church at Corinth that they should not either. Before, when Paul had opposed the Lord Jesus and His followers He only knew Jesus "after the flesh," meaning from a worldly point of view. It is only after a person is converted can he really know the Lord and the blessings which flow to believers as a result of His death upon the Cross.
In the verse which follows when Paul uses the words "old things have passed away" he is referring to the old way of evaluating people, that is, according to their outward appearance:
"Therefore if any man be in Christ, he is a new creature: old things are passed away; behold, all things are become new" (2 Cor.5:17).
At least that is the way I understand these verses. But I will admit that this passage is difficult to interpret and any other view on its meaning will be welcomed.
Thanks!