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Romans 6:1-7 clearly explains the meaning of baptism. There we are told that our old man was crucified with Christ and that in baptism we are buried with Christ into death. The old man is the mind that we had in our unconverted days that wanted to sin. That has been crucified with Christ.
We don't have to understand this first, before we live in the reality of it. We can just believe what God says. If God's Word says that our old man was crucified with Christ, then we believe it, just as surely as we believe God's Word when it says that Christ Himself was crucified on Calvary's hill. Both these truths are accepted by faith.
The old man and the flesh are not the same. The flesh is the Self-life within us, that opposes the will of God. We all have to carry this with us until our dying day. We could compare the flesh to a gang of robbers seeking to enter our house. The old man is like an unfaithful servant inside our house who constantly opened the door for the robbers to enter. It is the unfaithful servant who has now been killed. The robbers however are hale and hearty! But now we have a new servant, the new man, who seeks to keep the door shut, against these robbers.
In baptism, we testify to the death and burial of the old man (the desire to sin), and to being raised up with Christ so that we might henceforth "walk in newness of life" (Rom. 6:4).
The flood in Noah's day is a type of baptism too (1 Peter 3:20,21). The whole world was destroyed by God through that flood. Noah went through it in the ark and came out of it into a brand new world. The old world and everything in it were all buried under the flood. This is what we testify to in baptism as well - that our old relationship with the world (and that includes worldly fashions and worldly friends etc.,) has all now been cut off and that we are now coming out of the water into a brand new world.