A.C. Thiselton provides a good summary of the meaning of the Greek word sarx in his article in the New International Dictionary of New Testament Theology, Volume 1 (edited by Colin Brown, Grand Rapids: Zondervan, 1975, pages 671-682). The following are excerpts:

1. Paul uses sarx so frequently that only the most essential passages for its understanding can be discussed.

(a) As mere flesh..., sarx occurs only in 1 Cor. 15:39. Occasionally it denotes the human body...

(b) sarx also denotes relationships which have a merely temporary significance...

(c) sarx is used not only to indicate physical kinship; it can be used also generally in reference to what is human. Thus "the wise after the flesh" are the wise according to human standards (1 Cor 1:16...)...

(d) This leads to the use of the phrase kata sarka, according to the flesh. Even though Paul lives as a man "in the flesh", he did not carry on his fight in the light of human considerations ("after the flesh" 2 Cor. 10:2 f.; cf: 1:17)...

(e) The flesh, i.e. man's existence apart from God, has therefore a drie that is opposed to God. It not only occasions sin but also becomes entangled in it. Accordingly, Paul can draw up a catalogue of vices which he characterizes as "works of the flesh" or "desires of the flesh" (Gal 5:16, 19; cf. Rom 13:14)...

(f) On the one hand, Paul can say that the believer no longer lives in the flesh (Rom 7:5; 8:8 f.; Gal. 5:24). But on the other hand, as a believer, Paul still lives in the flesh (2 Cor. 10:3; Gal. 2:20; Phil. 1:22-24). The contradiction is resolved in Phil. 1:22-24. To be in the flesh is for him something that has been so vanquished, and no longer by one's own abilities. Even death, as departure and being with Christ, is much better. But for the sake of the future of the kingdom of God it is more important for him to remain in the flesh...

2. (a) The non-Pauline uses of sarx are, as might be expected, quite different...