What does it mean that Jesus fulfilled the law but did not abolish it? We hear that we are no longer under the law.  It even says so in Galatians:


  Galatians 3:23-25
  Before the coming of this faith, we were held in custody under the law, locked up until the faith that was to come would be revealed.  So the law was our guardian until Christ came that we might be justified by faith. Now that this faith has come, we are no longer under a guardian.


It seems 1 Corinthians 10:23 also supports this.

But yet, Jesus said that he did not come to abolish the law:


  Matthew 5:17-18
  “Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly I tell you, until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished."


What did Jesus mean that he did not come to abolish the law, but fulfill it, especially since it seems from later in the Bible that he did abolish the law.
I think part of the difficulty comes as a bit of confusion as to what is meant by the law and its application.   

When the law was introduced, it was never intended to be the instrument by which man would save himself, but it was the instrument through which man was to be saved.   In a very real sense, man is saved through the fulfilment (keeping) of the law.   When Christ fulfilled the law, he earned the privilege of life.  Those who are in Christ, likewise, share in his righteousness and are delcared righteous right along side him.   In that sense, the law has always stood and and will continue to stand.  

That being said, though, it's important to realize that we basically choose whether the law applies to us individually or not -- through grace, we can choose to be in Christ, or we can choose to stand on our own merits when we're judged.   In that sense, too, the law still stands.  

Paul talks a good deal about this in the book of Romans, and the whole book gives a much more complete (and competent :) ) version of an explanation that I'm about to...

I think it's important to realize the role of the law.  I'm going to jump arount a bit, but if we start with Romans 7:7-12, I think we can make a little more sense of this (emphasis mine):


  Romans 7:7What then shall we say? That the law is sin? By no means!
  Yet if it had not been for the law, I would not have known sin. For I
  would not have known what it is to covet if the law had not said, "You
  shall not covet." 8But sin, seizing an opportunity through the
  commandment, produced in me all kinds of covetousness. For apart from
  the law, sin lies dead. 9I was once alive apart from the law, but when
  the commandment came, sin came alive and I died. 10The very
  commandment that promised life proved to be death to me. 11For sin,
  seizing an opportunity through the commandment, deceived me and
  through it killed me. 12So the law is holy, and the commandment is
  holy and righteous and good.


There's a certain bit of irony when the law is introduced to human nature.   Being rebellious and sinful that we are, we latch on to the specifics as a means to found outlet for our repressed frustrated rebellions.   The law was never meant to save man, but rather was a contrast between the holiness of God and the depravity of man. The law offered life through its fulfilment and death through its violation.   This is a subtle but significant key, I think we somehow overlook when discussing grace, as this seems to be exactly how Jesus came to be our savior offering us hope for life...He earned that right by fulfiling the law.

That being said, there's still hope for us wretched sinners:


  Romans 7:4Likewise, my brothers, you also have died to the law through
  the body of Christ, so that you may belong to another, to him who has
  been raised from the dead, in order that we may bear fruit for God.
  5For while we were living in the flesh, our sinful passions, aroused
  by the law, were at work in our members to bear fruit for death. 6But
  now we are released from the law, having died to that which held us
  captive, so that we serve in the new way of the Spirit and not in the
  old way of the written code.


Looking at the block starting with v.7 we see that the law is not bad (in fact it's good) just because we use it as a springboard for badness.   The law is good, and it's been fulfilled by Christ.   He was the one who fulfilled it; not us.  

When we look on the Romams 8, Paul clarifies a bit more (Emphasis mine):


  Romans 8: 1 There is therefore now no condemnation for those who are
  in Christ Jesus. 2For the law of the Spirit of life has set you free
  in Christ Jesus from the law of sin and death. 3For God has done what
  the law, weakened by the flesh, could not do. By sending his own Son
  in the likeness of sinful flesh and for sin, he condemned sin in the
  flesh, 4in order that the righteous requirement of the law might be
  fulfilled in us, who walk not according to the flesh but according to
  the Spirit. 5For those who live according to the flesh set their minds
  on the things of the flesh, but those who live according to the Spirit
  set their minds on the things of the Spirit.


The righteous requirement of the law has been fulfilled in us (those who are in Christ Jesus) by virutue of us being in Christ.   The law has been fulfilled, there is, therefore no condemnation for those who are clothed with Christ's righteousness, but we should remember that those who live according to the spirit set their minds on spiritual things and will see to live in accordance with God's moral law.  Romans 6:15 reminds us of this, but note, (Paul does suggest again that we're not under the law, per se):


  Romans 6:15What then? Are we to sin because we are not under law but
  under grace? By no means! 16Do you not know that if you present
  yourselves to anyone as obedient slaves, you are slaves of the one
  whom you obey, either of sin, which leads to death, or of obedience,
  which leads to righteousness? 17But thanks be to God, that you who
  were once slaves of sin have become obedient from the heart to the
  standard of teaching to which you were committed, 18and, having been
  set free from sin, have become slaves of righteousness. 19 I am
  speaking in human terms, because of your natural limitations. For just
  as you once presented your members as slaves to impurity and to
  lawlessness leading to more lawlessness, so now present your members
  as slaves to righteousness leading to sanctification.


With his blood, Jesus purchased us from death.   Jesus was the fulfilment of the law; not us.  Because of that, we are bound as slaves of righteousness.   We're not going to fulfill the law in our own lives, and we're not expected to; but lest we use this as an excuse to sin without rein, we are supposed to set our sights on righteousness out of gratitude and duty to our Savior.  Those who understand the Gospel (the goodness of God, the wickedness of sin, the graciousness of God to provide a savior) will, naturally seek to please the God who save them and whom they serve, and the best way we can know what this is it to look to the law He's given us.

ETA: TL;DR:

When Jesus came to fulfil the law (i.e. obedience leads to life; disobedience leads to death) he did just that.   Notice He didn't say, that old law's no good, don't worry about it (i.e. abolish it); what he did, instead, was to come saying the law is good, and he is the fulfulment of it.