In our culture people mainly use the term unleavened bread when referring to the Lord's Supper. To most people yeast and leaven pretty much mean the same thing. There is a difference though that can really help one appreciate the Lord's Supper to an even greater extent.
Yeast can even refer to a single cell that is in the air. If you leave bread out the yeast cells will get in the bread and start a fermenting process that will cause the bread to rise. This can sometimes be called a starter bread. Apparently it takes a little work to get the starter bread going. You might have to moisten it and knead it for a while until the yeast has "leavened it".
In Old Testament times the Feast of Unleavened Bread was established. During this time the Israelites were to get leaven out of their homes. The "starter bread" and all breads they had made from it were to be thouroughly searched for and removed. Obviously there is no way they could get every yeast cell in the air out of their house. They however were to remove the starter bread and all breads made from it. There is symbolism in this since yeast is a "species of corruption". It ferments, or basically rots and corrupts to raise the bread.
In the gospels we find Jesus preparing for the Feast of Unleavened Bread with his disciples. On the first day of that feast he established the Lord's supper. This being the case, we know Jesus and the Jews with him had been sure to thoroughly remove leaven from the house they were in. Undoubtedly unleavened bread was used for the supper.Just as the Israelites searched to remove leaven from their homes as they were about to eat of the Paschal lamb, we should search diligently to remove the leaven of sin from our life as we accept Jesus as our Paschal lamb.
The concept of leaven and unleavened is used several other times in the New Testament.
The Leaven of The Pharisees and Sadducees
The Leaven of the Pharisees and Herod
Many believe the term Herod actually referred to the Herodians, a party like unto the Pharisees and Sadducees. The Herodians were known for ungodly sensual behavior. Some also have the theory that the Herodians were a "subset" of the Sadducees, or held the same beliefs as the Sadducees. That might explain why in parallel accounts Matthew says "Pharisees and Sadducees" and Mark says "Pharisees and Herod". Mark was just making a little more specific statement than Matthew was.
The Leaven of Malice and Evil
1Co 5:1-8 It is actually reported that there is sexual immorality among you, and of a kind that is not tolerated even among pagans, for a man has his father's wife. (2) And you are arrogant! Ought you not rather to mourn? Let him who has done this be removed from among you. (3) For though absent in body, I am present in spirit; and as if present, I have already pronounced judgment on the one who did such a thing. (4) When you are assembled in the name of the Lord Jesus and my spirit is present, with the power of our Lord Jesus, (5) you are to deliver this man to Satan for the destruction of the flesh, so that his spirit may be saved in the day of the Lord. (6) Your boasting is not good. Do you not know that a little leaven leavens the whole lump? (7) Cleanse out the old leaven that you may be a new lump, as you really are unleavened. For Christ, our Passover lamb, has been sacrificed. (8) Let us therefore celebrate the festival, not with the old leaven, the leaven of malice and evil, but with the unleavened bread of sincerity and truth.
The Leaven of False Doctrine