In Matthew 5:17, Jesus says he didn't come to abolish the law or Prophets but to fulfill them. What does it mean to "fulfill the law"?
Some possible meanings:
I've always thought "fulfill the Law" referred to the death and resurrection of Christ. In the sense that, it was the Law that condemns us, and it is faith that justifies us. In other words, God had always intended that the Law be perfected/completed/revealed in Christ.
That said, in studying this question I came across a few verses that have me wondering if there is more to what "fulfill the law" means than what I've thought...
I'll need to chew on that and am looking forward to somebody else shedding light.
To fulfill the law means to complete it in every aspect. The Greek word is πληρῶσαι (lexical form πληρόω). Τελειόω is a synonym; it has a sense of bringing something to completeness in its end, in its finality. Πληρόω has the sense of to complete something in fullness. And this sense indeed applies to Jesus' fulfillment of the law: it was not a meager point-by-point legalistic accomplishment. It was a perfect, full, absolute, complete accomplishment.
This fullness of completeness is due to the fundamental nature of the law. Let's stay in the book of Matthew. If we are to talk about what law Jesus fulfilled, we must go to the passage in which he himself expresses what the law is.
In its most fundamental aspect, then, Jesus' keeping of the law was his perfect, passionate, perpetual love of the Father, and his gracious, humble, compassionate love to sinners. This is certainly played out in a detailed way, as expressed in Galactic Cowboy's answer, but let us not loose sight of the fact that everything Jesus did was out of love for the Father.
It's all about communion.
Jesus fulfills the prophecies which are hidden in the law.
The law says a clean animal is one which ruminates (meditates on the word) and that meditation produces a separated (holy) walk.
Jesus fulfills the prophecy contained in the law of the leper. His Father forsook him (shaved head), his prayer (remove this cup) was not answered (covered lip). He bore our sin completely (leper covered completely with sin) and became our high priest (ceremony of cleansing like the one for priests).
He fulfills the law of murdering by killing without becoming culpable of murder by laying down his own life in love for us.
He fulfills all the law of sacrifices since they are all types of his sacrifice.
He fulfills the law of lying by lying without becoming culpable of sin by speaking in parables.
He 'does not honor his parents' when he was twelve without sinning by 'being about His Father's business'.
He comes as the thief in the night.
All the law has a shadow which is fulfilled in Christ.