St. Paul writes, “Now I rejoice in my sufferings for your sake, and in my flesh I complete what is lacking in Christ’s afflictions for the sake of his body, that is, the church.”
What this doesn’t mean is that Christ’s death on the cross was incomplete with regard to redeeming the whole human race and reuniting it with God. There is nothing that anyone can add to the infinite value of the cross.
However, there is an aspect of Christ’s suffering that is not complete—namely, the application of the merits of Christ’s Passion to individual souls. It is this aspect of redemption (the subjective dimension) of which Paul speaks. According to A Catholic Commentary on Holy Scripture, “[Paul’s sufferings] are the vehicle for conveying the Passion to the hearts and souls of men, and in this way they bring completeness to the Passion in an external way.” This is not the only time that Paul refers to his own sufferings in the service of Christ as Christ’s afflictions in his own flesh (see 2 Cor. 1:5, 4:10; Phil. 3:10).
Another opinion is that of St. Augustine, who regarded “Christ’s afflictions” as the suffering of the mystical body: “Thou [member of Christ’s Body] sufferest so much as was to be contributed out of thy sufferings to the whole sufferings of Christ, that hath suffered in our Head, and doth suffer in his members, that is, in our own selves (Enarr. in Ps. 62:4).” The idea here is that Christ’s Passion is continued in the members of his mystical body as they unite their sufferings to Jesus, the head.
Either opinion is a legitimate way to read Colossians 1:24, although many prefer the first.