Jesus likened the hypocrisy of the Pharisees and Sadducees to leaven.

Mat 16:11  How is it you do not understand that I did not speak to you concerning bread?—but to beware of the leaven of the Pharisees and Sadducees.”
Mat 16:12  Then they understood that He did not tell them to beware of the leaven of bread, but of the doctrine of the Pharisees and Sadducees.

Luk 12:1  In the meantime, when an innumerable multitude of people had gathered together, so that they trampled one another, He began to say to His disciples first of all, “Beware of the leaven of the Pharisees, which is hypocrisy.

Paul spoke of leaven being like malice and wickedness.

1Co 5:6  Your glorying is not good. Do you not know that a little leaven leavens the whole lump?
1Co 5:7  Therefore purge out the old leaven, that you may be a new lump, since you truly are unleavened. For indeed Christ, our Passover, was sacrificed for us.
1Co 5:8  Therefore let us keep the feast, not with old leaven, nor with the leaven of malice and wickedness, but with the unleavened bread of sincerity and truth.

Paul also likened not obeying the truth to leaven.

Gal 5:7  You ran well. Who hindered you from obeying the truth?
Gal 5:8  This persuasion does not come from Him who calls you.
Gal 5:9  A little leaven leavens the whole lump.

So leaven pictures hypocrisy, malice, wickedness, and disobedience to God.  This can be summed up in one word, sin.  But why did God choose leaven as something to picture sin?  Perhaps, because a little leaven leavens the whole lump is similar to the way in which sin can so easily spread.

So, leaven can picture sin.  But leaven can also picture something else.

The parable of the leaven likens leaven to the kingdom of God.

Mat 13:33  Another parable He spoke to them: “The kingdom of heaven is like leaven, which a woman took and hid in three measures of meal till it was all leavened.”

Luk 13:20  And again He said, “To what shall I liken the kingdom of God?
Luk 13:21  It is like leaven, which a woman took and hid in three measures of meal till it was all leavened.”

Just as leaven spreads throughout the whole lump of dough so the kingdom of God will spread and grow. 

Conclusion

The same property of leaven—its ability to puff up and expand—is used to picture both sin and the kingdom of God.