1 CORINTHIANS 11:23 For I received from the Lord that which I also delivered to you, that the Lord Jesus in the night in which He was betrayed took bread;
 24 and when He had given thanks, He broke it, and said, "This is My body, which is for you; do this in remembrance of Me."
 25 In the same way He took the cup also, after supper, saying, "This cup is the new covenant in My blood; do this, as often as you drink it, in remembrance of Me."

We also find an EXAMPLE of the church participating in this memorial from the beginning in Jerusalem:

ACTS 2:42 And they were continually devoting themselves to the apostles' teaching and to fellowship, to the breaking of bread and to prayer.

[NOTE: The definite article “the” sets this aside as a special observance “THE breaking of bread,” as opposed to the general “breaking of bread,” or hospitality (v.46), which the Christians also shared, but is noted separately from the acts of assembled worship found in v.42. For “breaking of bread” = communion, see 1 Corinthians 10:16.]

From the example of the church in Troas, we note that communion was observed in the assembly on the first day of the week:

ACTS 20:7  And on the first day of the week, when we were gathered together to break bread, Paul began talking to them, intending to depart the next day, and he prolonged his message until midnight.

We also have more general indicators of Christians assembling regularly on the first day of the week. This seemed a convenient time in Paul’s view for Christians to take up collection for the needy saints in Judea:

1 CORINTHIANS 16:2 On the first day of every week let each one of you put aside and save, as he may prosper, that no collections be made when I come.

So, that’s the “when” and “how often” of it, but why should we partake of Christ’s memorial every week?

1 CORINTHIANS 11:25 In the same way He took the cup also, after supper, saying, "This cup is the new covenant in My blood; do this, as often as you drink it, in remembrance of Me."
  26 For as often as you eat this bread and drink the cup, you proclaim the Lord's death until He comes.

As Christians, we live in a covenant relationship with God, and Christ’s blood was the sacrifice that initiated that relationship.

1 JOHN 4:10 In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins.

We take that blood upon ourselves when we are baptized into Christ. “Or do you not know that all of us who have been baptized into Christ Jesus have been baptized into His death?” (Romans 6:3). Christ’s memorial stands as a witness, or memorial. Such witnesses have always been a feature of covenant-making in the Bible, serving as reminders to the participants of what they had done.

MATTHEW 26:28 “ …for this is My blood of the covenant, which is poured out for many for forgiveness of sins.”

The Lord’s Supper is a constant reminder of the new covenant and its place in the death of Christ. His death is central to forgiveness; every time we partake, it is a pledge that we, whose covenant was established at such cost, will live in a fitting manner.