Romans 8:1-8

There is therefore now no condemnation to those who are in Christ Jesus, who do not walk according to the flesh, but according to the spirit. For the law of the Spirit of Life in Christ Jesus has made me free from the law of sin and death.  

The “Therefore” is referring to the previous chapter, where Paul talks about the law being unable to justify, and that the righteousness of Jesus has made us free from the law, similar to a woman whose husband dies and she is now free to marry again. The law is dead and we have been joined with Christ, where the “law” now becomes our voluntary love for God instead of our obligatory requirement. Chapter 7 ends with Paul being thankful that Christ has freed us from the struggle of our warring members so that while we are still at war with our flesh, we no longer bear the consequences of it.

Therefore, because we are “widowed” from the law and our new marriage is based only on the devotion of our hearts rather than our actions, we cannot be condemned because Christ cannot be condemned. Our old husband was there for the sole purpose of condemnation, but our new husband is there for justification. The sign of our being “in Christ” is if we follow the new law, which is simply loving God. John 14:21 says, “He who has my commandments and obeys them, it is he who loves me.” What commandments? In Matthew 22:36-40, Jesus states that all the law and the prophets hang on first love for God, and secondly love for people. To prove love for God, you need to follow His commandment, which is loving Him with all your heart, soul and mind. It is very simple, but not as cut and dried as the law.

Even though it is much less openly apparent if someone is following the new law instead of the old law, there are several “heart checks” you can do. Matthew 6:21 says that, “Where your treasure is, there your heart will be also.” One practical way to see if you love God is to see where your “treasure” is. What do you value? Where does your time go? Where does your money go? What things do you guard as the most sacred with both possessions and time? If an honest evaluation shows that you guard your time with the LORD and that you demonstrate through giving that your finances are available for God’s use, then you can be pretty sure that you do love God in the way that Jesus was talking about.

Understanding what it means to walk according to the Spirit instead of the flesh is an extremely important concept to get a grasp on, but it is also difficult due to its subjective nature.

For what the law could not do in that it was weak through the flesh, God did by sending His own Son in the likeness of sinful flesh, on account of sin.

There was nothing really wrong with the concept of the original law. It could theoretically lead to salvation because if perfectly kept, one would be perfectly righteous before God. The one thing that makes the law weak is that we in our fleshly wickedness and weakness are utterly unable to follow it. The intent of the original law was not to make us righteous, but to prove our unrighteousness. Jesus came as a man to atone us because we were unable to do so for ourselves. Had Jesus committed even a single sin, He would have been disqualified from being our atonement because He would have suffered the wrath of God for His own unrighteousness. Because He was absolutely perfect and blameless according to the law of God (keeping in mind the ridiculous extra rules the Pharisees created are NOT the law of God), he did not have to suffer the wrath of God; however, the option was open to Him to suffer the wrath of God on our behalf. Not the wrath of a hateful God against a creation that He was no longer happy with, but the wrath of a righteous and holy God against the sin which had caused our eternal separation from the creation He loved. Sin cannot come into the presence of a holy God any more than darkness can creep into a well-lit room. Once the light is turned on, the darkness is banished out of the room and cannot enter the room unless the light goes away. It’s the same with our sin. Because our NATURE is sinful, we are bound to it like yeast in a loaf of bread. It cannot be extracted piece by piece leaving the loaf intact as it was before. For us to really enter God’s presence, we would be destroyed along with the darkness inside us. That’s why it is impossible for us to go to heaven on our own power.

He condemned sin in the flesh that the righteous requirement of the law might be fulfilled in us who do not walk according to the flesh but according to the Spirit.

The Greek word for “in” is en, which just like in English is a preposition representing a position. It is not an action we perform. It is not fulfilled because of us or through us, but inside of us by the work Christ did replacing our righteousness with His. The Greek for “walk” is peripateō which basically means to make your way or navigate. It doesn’t mean our works are spiritual and holy, but rather that the decisions we make are guided by our spirit instead of our flesh. We “walk according to the Spirit” by using the Spirit as our compass instead of the law or any other fleshly thing we would normally use as an indicator of what direction we should take.

For those who walk according to the flesh set their minds on the things of the flesh, but those who walk according to the Spirit, the things of the Spirit.

The mindset we are in when the Spirit is our compass instead of earthly things is that in every situation we evaluate how our actions relate to our walk with Christ and we evaluate His approval/disapproval. We are always concerned with the eternal implications and not concerned with fulfilling our flesh or the impact of our actions on earthly things such as popularity, position, and comfort.

For to be carnally minded is death, but to be spiritually minded is life and peace.

When we take the burden on ourselves, the best we can do is fail with good intentions. Death. We cannot reach the requirement. If we devote ourselves to fulfilling our popularity, position, and comfort, we give our precious time to things that will not endure past death, if they even make it that far. To be spiritually minded, we place our hope beyond this world, so nothing that goes wrong in this world can shake our eternal perspective. There is nothing bad that can happen here that will endure through eternity besides our decision to deny the gift of salvation our entire lives.

Because the carnal mind is enmity against God and is not subject to the law of God, nor indeed can be,

The law of God is love for God. The text seems to imply that we are carnally minded even if we are trying to follow the law and please God if we are doing so in our own righteousness, not motivated by love for God but by a desire for self-righteous salvation.   

So what?

We have to have a change in our thinking when it comes to pleasing God. The law was not created so that if we follow it good enough, God would be happy with us. There is no "good enough" There is perfect and there is failure, and if you're old enough to know how to read what I'm typing, then you've had plenty of time to prove to yourself that you are completely unable to be perfect. We can say that we are saved by grace all day long but when we really understand that concept, a significant change occurs in the way we go about our lives. 
Jesus did not do away with this old law, but fulfilled it (Matthew 5:17). Nothing changed in the requirement. That is still perfection. 
As Christians we still try to do good enough, even though we should understand that we either pass or fail, and we have already failed. 
As I went over before, there is a new commandment, one that is one a higher level than the old law, because if the new commandment is followed, the resulting righteousness will be greater than the righteousness of someone who was excellent at following the old law. See Matthew 5:20 where Jesus says that unless someone is MORE righteous than the Sribes and Pharisees, he will not enter the Kingdom of Heaven. The Scribes and Pharisees literally made a career of legalism and understanding the law and creating ways to ensure that they follow it by making extra rules. The implication (their righteousness must be EXCEEDED) is that even this was not good enough.
The new and greater commandment that is sufficient for pleasing God is the commandment to love God with all your heart, soul, mind, and strength. This is not a little thing. For one, this is the one thing we need to get because it is the key to how we are to conduct ourselves for the rest of this life. We have to get it. The alternative is failure.
The second reason this idea is so big is because it is so much more difficult than it is often made to sound. Do you REALLY love God with all your heart? Where is the evidence? What are your motivations? Do you hunger and thirst for God? Are you willing to give up the things you want most in the world if God has other plans? When there is a quiet spot in your day are you meditating on God?
I know when I was dating Heather that when I wasn't with her, I was thinking about her because no matter what I was doing, I didn't want to be doing it as much as I wanted to be with her. My mind, my heart, my efforts, all of it was tied to trying to be with her again.
That's the idea, though the application is a lot more difficult. We can never depend on hormones or feelings to propel us toward God. More often than not our chemistry and circumstances are pulling our hearts AWAY. We must master that as it says in 2 Corinthians 10:5. We must come to the place where we subject our feelings to do what we know in our heads is right. Our minds are the rudder and our emotions are the sails. Sails with no rudder will bring you places you don't want to go and a rudder with no sails will leave you stranded.
You must set your mind and will to turn your heart to God, and then only by sumitting your will to the Spirit will you be able to carry out subjecting your emotions toward loving God with everything you are. 
That is what I believe the key is to "not walking according to the flesh, but according to the Spirit".