Jamieson, Fausset and Brown on 1 Corinthians 11:25

25. when he had supped--Greek, "after the eating of supper," namely, the Passover supper which preceded the Lord's Supper, as the love-feast did subsequently. Therefore, you Corinthians ought to separate common meals from the Lord's Supper [Bengel].

the new testament--or "covenant." The cup is the parchment-deed, as it were, on which My new covenant, or last will is written and sealed, making over to you all blessings here and hereafter.

in my blood--ratified by MY blood: "not by the blood of goats and calves" (He 9:12).

as oft as--Greek, "as many times soever": implying that it is an ordinance often to be partaken of.

in remembrance of me--Luke (Lu 22:19) expresses this, which is understood by Matthew and Mark. Paul twice records it (1Co 11:24 and here) as suiting his purpose. The old sacrifices brought sins continually to remembrance (He 10:1, 3). The Lord's Supper brings to remembrance Christ and His sacrifice once for all for the full and final remission of sins.