I always wanted to know that as well, and prayed and prayed, and the Lord blessed me!
Walking after/in the Spirit is leaning on the Spirit for empowerment in order to please God in a real Christian life.
Romans 7:6 says that it is not enough to know we must serve God. We are shown there is serving God after the letter of the law, and serving God
in newness of
Spirit.
Romans 7:6 KJV But now we are delivered from the law, that being dead wherein we were held; that we should serve in newness of spirit, and not in the oldness of the letter.
Paul then explains how there is a struggle when a person wants to do the will of God but cannot find it in his/her flesh to perform that will of God.
Romans 7:18 KJV For I know that in me (that is, in my flesh,) dwelleth no good thing: for to will is present with me; but how to perform that which is good I find not.
This frustration is due to walking in the flesh to serve God. That is what oldness of the letter meant. All they had in Old Testament times was fleshly ability, since God did not empower them with Spirit baptism in those days. And they failed miserably. But believers today also fail if they try to serve God today using fleshly effort alone. You can have God's Spirit and power in you, but if you do not know it's there to empower you as a result of your prayers, you will continue to try your human best to do His will and fail. That is why
Romans 6 repeats the question, "Know ye not?"
This is called walking after the flesh. Notice how Paul uses the term FLESH in these verses:
Romans 7:25-8:1 KJV I thank God through Jesus Christ our Lord. So then with the mind I myself serve the law of God; but with the flesh the law of sin. (8:1) There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit.
Since serving God with the mind is useless alone, since our efforts to use the FLESH to fulfill that will of God do not work (compare 7:18 with 7:25), Paul says to STOP WALKING AFTER THAT FLESH. When we use fleshly effort to fulfill the will of God that our minds want to do, we fail and fail and fail. Paul summarized verse 18 in verse 25. So Paul says to not walk after that flesh that we try to use to fulfill what our minds desire to do. Instead, walk after the Spirit.
And this brings us to your question. Walking after the flesh is trying in human effort alone to do the will of God that your mind wills to do. But walking after the Spirit is having faith in the Spirit of God to quicken or empower you to do the will of God. It is basically what
Romans 6:13 is telling us to do:
Romans 6:13 KJV Neither yield ye your members as instruments of unrighteousness unto sin: but yield yourselves unto God, as those that are alive from the dead, and your members as instruments of righteousness unto God.
Actually pray for God to give you power to do His will, instead of just going ahead and trying to do His will in your human abilities. Many do not even realize THEY CAN ASK GOD TO GIVE THEM POWER to do His will! This is called walking after the Spirit.
Walking after something is living by it. So we are either living by our fleshly will power (walking after the flesh) or by the Spirit (walking after/in the Spirit).
When we resort to God's Spirit to give us power, as we present ourselves in faith for Him to do so, HE WILL ANSWER! But it takes time to really get this in our blood and always lean on the Spirit to do His will, since it is so easy to slip back into natural instinct and try doing His will in human effort alone without reliance on His Spirit.
There's much more, but that is a start!