The cup of blessing which we bless, is it not the communion (κοινωνία) of the blood of Christ? The bread which we break, is it not the communion (κοινωνία) of the body of Christ? (1 Cor 10:16)
In his teaching on the nature of the Eucharist/Lord’s Supper, Paul informs the Corinthian church that, when partaking of the bread and wine, such is a “communion” with the body and blood of Christ. The term translated as “communion” is the Greek term κοινωνία and means "participation/sharing"; such is consistent with the theology of Col 1:24 which supports a participatory model of atonement. When defining the term, BDAG discusses its relationship to the meaning of 1 Cor 10:16:
3. abstr. for concr. sign of fellowship, proof of brotherly unity, even gift, contribution (Lev 5:21; ins of Asia Minor: κ.=‘subsidy’ [Rdm.2 10]) Ro 15:26 (s. 1 above). Under this head we may perh. classify κοινωνία τ. αἵματος (σώματος) τοῦ Χριστοῦ a means for attaining a close relationship with the blood (body) of Christ 1 Cor 10:16ab (s. 4 below).
4. participation, sharing τινός in someth. (Appian, Bell. Civ. 1, 67 §306 κ. τῶν παρόντων=in the present undertakings; 5, 71 §299 κ. τῆς ἀρχῆς in the rule; Polyaenus 6, 7, 2 κ. τοῦ μιάσματος in the foul deed; Maximus Tyr. 19, 3b τῆς ἀρετῆς; Synes., Kgdm. 13 p. 12c. κ. τῶν ἔργων=in the deeds of others; Wsd 8:18; Jos., Ant. 2, 62) ὅπως ἡ κ. τῆς πίστεώς σου ἐνεργὴς γένηται that your participation in the faith may be made known through your deeds Phlm 6. γνῶναι κοινωνίαν παθημάτων αὐτοῦ become aware of sharing his sufferings Phil 3:10. ἡ κ. τῆς διακονίας τῆς εἰς τοὺς ἁγίους taking part in the relief of God’s people 2 Cor 8:4. Perh. this is the place for 1 Cor 1:9 (s. 1 above); 2 Cor 13:13 ( participation in the Holy Spirit: Ltzm., Kümmel in appendix to Ltzm. comm., Windisch, Seesemann [s. below] 70; Goodsp., Probs. 169f; s. 1 above.—Cp. τοῦ ἁγίου πνεύματος κ. of ecstasy Did., Gen. 230, 16); 1 Cor 10:16 (participation in the blood [body] of Christ. So ASchlatter, Pls der Bote Jesu ’34, 295f et al.; s. 3 above. But perh. here κοινωνία w. gen. means the common possession or enjoyment of someth. [Diod. S. 8, 5, 1 ἀγελῶν κ.= of the flocks; Maximus Tyr. 19, 3b ἐπὶ κοινωνίᾳ τῆς ἀρετῆς=for the common possession of excellence; Diog. L. 7, 124; Synes., Kgdm. 20 p. 24b; Hierocles 6, 428: we are to choose the best man as friend and unite ourselves with him πρὸς τὴν τῶν ἀρετῶν κοινωνίαν=for the common possession or enjoyment of virtues; 7, 429 τῶν καλῶν τὴν κ.]. Then 1 Cor 10:16 would be: Do not the cup and the bread mean the common partaking of the body and blood of Christ? After all, we all partake of one and the same bread). Eph 3:9 v.l. (for οἰκονομία)—JCampbell, Κοινωνία and Its Cognates in the NT: JBL 51, ’32, 352-80; EGroenewald, Κοινωνία (gemeenskap) bij Pls, diss. Amst. ’32; HSeesemann, D. Begriff Κοινωνία im NT ’33; PEndenburg, Koinoonia … bij de Grieken in den klass. tijd ’37; HFord, The NT Conception of Fellowship: Shane Quarterly 6, ’45, 188-215; GJourdan, Κοινωνία in 1 Cor 10:16: JBL 67, ’48, 111-24; KNickle, The Collection, A Study in Paul’s Strategy, ’66.—EDNT additional bibl. S. also RAC IX 1100-1145.—DELG s.v. κοινός. M-M. TW. Sv.
Furthermore, as one commentator noted, the theme of this concept is that of the “centrality of divine grace and of eschatology that makes corporate solidarity with Christ the basis of this communal participation and unit” (Anthony C. Thiselton, The First Epistle to the Corinthians [NIGTC; Grand Rapids, Mich.: Eerdmans, 2000] 763).
This is another important aspect that should inform anyone’s theology of the Lord Supper as well as ones approach thereto—we are engaging in a spiritual participation with the body and blood (and, as a result, sacrificial death) of Christ during the celebration of the sacrament.