What the Apostle means by flesh in this passage, and what the lust of the flesh is.
Wherefore in this passage we ought to take “flesh” as meaning not man, i.e., his material substance, but the carnal will and evil desires, just as “spirit” does not mean anything material, but the good and spiritual desires of the soul: a meaning which the blessed Apostle has clearly given just before, where he begins: “But I say, walk in the spirit, and you shall not fulfil the desires of the flesh; for the flesh lusts against the spirit and the spirit against the flesh: but these are contrary the one to the other, that you may not do what you would.” And since these two; viz., the desires of the flesh and of the spirit co-exist in one and the same man, there arises an internal warfare daily carried on within us, while the lust of the flesh which rushes blindly towards sin, revels in those delights which are connected with present ease. And on the other hand the desire of the spirit is opposed to these, and wishes to be entirely absorbed in spiritual efforts, so that it actually wants to be rid of even the necessary uses of the flesh, longing to be so constantly taken up with these things as to desire to have no share of anxiety about the weakness of the flesh. The flesh delights in wantonness and lust: the spirit does not even tolerate natural desires. The one wants to have plenty of sleep, and to be satiated with food: the other is nourished with vigils and fasting, so as to be unwilling even to admit of sleep and food for the needful purposes of life. The one longs to be enriched with plenty of everything, the other is satisfied even without the possession of a daily supply of scanty food. The one seeks to look sleek by means of baths, and to be surrounded every day by crowds of flatterers, the other delights in dirt and filth, and the solitude of the inaccessible desert, and dreads the approach of all mortal men. The one lives on the esteem and applause of men, the other glories in injuries offered to it, and in persecutions.
Source: Conferences (New Advent)