I was reading Romans 8 today and pondering that question. The understanding I have come to, and I'm not sure from who or where, is that "flesh" refers to the conditioned body- the entropy aspect of our physical selves which embeds habit, trauma, prejudice and addiction. It is also the aspect of ourselves which lives blindly for ourselves alone- what is called, in Jewish thought, the nefesh behema- the animal soul (eg. Tanya 1, R'Shneur Zalman of Liadi). This is not to disparage animals, who live beautifully within God's plan as innocent aspects of the image of His glory (Catechism of the Catholic Church). Human beings, however, are not intended to serve God's plan by blindly following the dictates of our physical conditioning. Having had a divine soul blown into us (Genesis 2:6) we reflect the image of God in a special way (Genesis 1:27). It is our choice to be continually open to this Spirit which was, is, and may be blown in to us, moving where it wills (John 3:8) and opening our eyes to ever new things.
In Romans 8 Paul says that "those who live according to the flesh have their minds set on what the flesh desires, but those who live in accordance with the Spirit have their minds set on what the Spirit desires (8:5)". The flesh desires the increase of itself, which is all that cells, neurons and ATP know how to do. In Chinese folklore this aspect of the self is called the "po", or "physical soul" and is said to die with the body. Thus Paul says that "if you live according to the flesh, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live (8:13)". The misdeeds of the body are living for what dies and in the momentum of the flesh instead of the ever new light of the Spirit, which brings life.
What is living? It is growth, vitality, vividness, wisdom, consciousness, expansion. When we live to the Spirit we are truly alive, and life is a bracing, challenging, heartbreaking and heart expanding way of never ending growth, or reaching forward into the future in a God-ruled becoming: "And if the Spirit of the one who raised Jesus from the dead is living in you, he who raised the Christ from the dead will also give life to your mortal bodies because of the Spirit who lives in you (8:11)." This is a spirit which conquers death, entropy, and the dead end. To return to Chinese folklore, this is the yang of new life, not the yin of stagnancy and finality (David Gelernter, Judaism: A Way of Being). It is not a return to primordial simplicity but an expansion into greater, more versatile complexity. As Spinoza said, the more complex our ability to feel and respond, the greater our perfection (Ethics p. 4, Appendix:27). Entropy and habituation limit our ability to feel, act and think for ourselves and thus in Spinoza's thought make us slaves of our passions and of the external world, or in other words, of the flesh. The way of the Spirit is a paradoxical way where the more we surrender to God and the gift of the Spirit within the more individuated and powerful we become, filled with a life that is simultaneously not our life but that gives us our life , as Jesus said (Matthew 10:39): "Whoever loses his life for my sake will find it." We "lose our lives" by continually dying to the flesh and living towards the Spirit.