What is the blasphemy of the Holy Spirit?

In order to answer your question it is necessary that we first understand the meaning of Blaspheme, and then see how that applies to the Holy Spirit, and ultimately who or what the Holy Spirit is in accordance with the Bible.

All Scripture is quoted from the King James translation

According to Merriam Webster's dictionary the verb blaspheme has two connotations. It can be either transitive are intransitive. Transitive: BLASPHE'ME, v.t. is [Gr. The first syllable is the same as in blame, blasme,

denoting injury; L. loedo,   loesus; The last syllable is the Greek.,to
speak.]

    1. To speak of the Supreme Being in terms of impious irreverence; 
    to revile or speak reproachfully of God, or the Holy Spirit. 
    As in: 1 Kings 21. Mark 3.

    2. To speak evil of; to utter abuse or calumny against; to speak 
    reproachfully of.

Intransitive:

BLASPHE'ME, v.i. To utter blasphemy.

Matthew 12:32 And whosoever speaketh a word against the Son of man, it shall be forgiven him: but whosoever speaketh against the Holy Ghost, it shall not be forgiven him, neither in this world, neither in the world to come.

    1. To arrogate the prerogatives of God.

    This man blasphemeth. Who can forgive sins but God? Mathew.9.
     Mark 2.

As we see above in Matthew chapter 12 verse 31 the verb blaspheme is used in the intransitive form. I have included the definition of the word arrogate since it is not a common word.

AR'ROGATE, v.t. [L. arrogo, of ad and rogo.]

To assume, demand or challenge more than is proper; to make undue
claims, from vanity or false pretensions to right or merit; as, the Pope
arrogated dominion over kings.

To this point. What we have done is to define these two verbs. Blaspheme and arrogate, now we must determine what Jesus meant in his statement. To do this we need to look at the setting, and to whom he was talking at the time.

Matthew 12:24 through 27 But when the Pharisees heard it, they said, This fellow doth not cast out devils, but by Beelzebub the prince of the devils. And Jesus knew their thoughts, and said unto them, Every kingdom divided against itself is brought to desolation; and every city or house divided against itself shall not stand: And if Satan cast out Satan, he is divided against himself; how shall then his kingdom stand? And if I by Beelzebub cast out devils, by whom do your children cast them out? therefore they shall be your judges. Here we find that Jesus is accused of being inhabited by Satan and it is the power of Satan within him that he uses to cast out Demons. Jesus answers them by saying that they are not making sense, why would Satan want to cast out other Devils, since that would be exactly opposite of his goals.

Then we come to the part in which is your primary interest:

Matthew 12:28 and 29 But if I cast out devils by the Spirit of God, then the kingdom of God is come unto you. Or else how can one enter into a strong man's house, and spoil his goods, except he first bind the strong man? and then he will spoil his house. We need to break this down to see just what Jesus meant:

1.The strong man referes to God, and the strong man's house refers to Heaven.

2.In order to be able to have the power to be supreme it would be necessary
to defeat God and then he would be able to dictate how God's power would be 
used.

3.In saying that the sins of man including other blasphemies (not including
blasphemies against God) shall be forgiven, however, Blasphemies against 
the Holy Ghost will never be forgiven, and whoever does is in danger of 
eternal punishment.

Finally; who or what is the Holy Ghost (Spirit)?

So let's see if we can determine Biblically who this Holy Ghost is:

Matthew 1:18 Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost.

Matthew 3:11 I indeed baptize you with water unto repentance: but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire:

Mark 12:36 For David himself said by the Holy Ghost, The LORD said to my Lord, Sit thou on my right hand, till I make thine enemies thy footstool.

Mark 13:11 But when they shall lead you, and deliver you up, take no thought beforehand what ye shall speak, neither do ye premeditate: but whatsoever shall be given you in that hour, that speak ye: for it is not ye that speak, but the Holy Ghost.

Luke 1:15 For he shall be great in the sight of the Lord, and shall drink neither wine nor strong drink; and he shall be filled with the Holy Ghost, even from his mother's womb.

Luke 1:35 And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.

Luke 12:12 For the Holy Ghost shall teach you in the same hour what ye ought to say.

John 20:22 And when he had said this, he breathed on them, and saith unto them, Receive ye the Holy Ghost:

When we closely examine these we see one common thread, and that is that the Holy Ghost is some sort of superior power, authority, and intelligence which comes from God.

Going back to our definition of arrogate, it appears that in this case Jesus is talking about the Scribes, ascribing the nature of God to Satan, and therefore at least in this case, The Holy Spirit refers to the nature of God, and to deny or ascribe that to any other than God is blaspheming the Holy Ghost.

Mark's answer is certainly true, but there is more to the story. To find out, we really need to read this passage in context.

Matthew 12:22-37(NIV):

22 Then they brought him a demon-possessed man who was blind and mute, and Jesus healed him, so that he could both talk and see. 23 All the people were astonished and said, “Could this be the Son of David?”
24 But when the Pharisees heard this, they said, “It is only by Beelzebul, the prince of demons, that this fellow drives out demons.”
25 Jesus knew their thoughts and said to them, “Every kingdom divided against itself will be ruined, and every city or household divided against itself will not stand.
26 If Satan drives out Satan, he is divided against himself. How then can his kingdom stand?
27 And if I drive out demons by Beelzebul, by whom do your people drive them out? So then, they will be your judges.
28 But if it is by the Spirit of God that I drive out demons, then the kingdom of God has come upon you.
29 “Or again, how can anyone enter a strong man’s house and carry off his possessions unless he first ties up the strong man? Then he can plunder his house.
30 “Whoever is not with me is against me, and whoever does not gather with me scatters.
31 And so I tell you, every kind of sin and slander can be forgiven, but blasphemy against the Spirit will not be forgiven.
32 Anyone who speaks a word against the Son of Man will be forgiven, but anyone who speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come.
33 “Make a tree good and its fruit will be good, or make a tree bad and its fruit will be bad, for a tree is recognized by its fruit.
34 You brood of vipers, how can you who are evil say anything good? For the mouth speaks what the heart is full of.
35 A good man brings good things out of the good stored up in him, and an evil man brings evil things out of the evil stored up in him.
36 But I tell you that everyone will have to give account on the day of judgment for every empty word they have spoken.
37 For by your words you will be acquitted, and by your words you will be condemned.”

So if we look at the context of this, we can clearly see what provoked Jesus to say this. It was when the Pharisees were calling the work of the Spirit the work of beelzebul. When Jesus performed the miracle, they denied it as being from God. Jesus explained that 1. it was really stupid to say that the devil would cure somebody since he's the one that cursed them. and 2. The Holy Spirit is the one who was giving Him the power to drive out the deamons, and calling the work of the Holy Spirit evil, is the worst thing you could ever do and you will be held accountable for it. He was making it clear that they had crossed the line from just discrediting Him (they thought he was only a man, so it was nothing for them to speak against Him), to blaspheming the Spirit of God.

So in context, we see that blasphemy against the Holy Spirit is when the Holy Spirit is doing something miraculous and you turn around and say that it is of the devil.

To understand blasphemy against the Spirit, one first needs to understand what blasphemy is. The Catechism of the Catholic Church defines it thusly:

Blasphemy is directly opposed to the second commandment. It consists in uttering against God - inwardly or outwardly - words of hatred, reproach, or defiance; in speaking ill of God; in failing in respect toward him in one's speech; in misusing God's name. St. James condemns those "who blaspheme that honorable name [of Jesus] by which you are called." The prohibition of blasphemy extends to language against Christ's Church, the saints, and sacred things. It is also blasphemous to make use of God's name to cover up criminal practices, to reduce peoples to servitude, to torture persons or put them to death. The misuse of God's name to commit a crime can provoke others to repudiate religion.

Basically, it's going against God's will and treating Him with anything less than the respect and love He deserves.

The Holy Spirit, being one part of the Holy Trinity, is what confers the gift of grace upon people. So, to blaspheme against the Spirit is to willfully reject God's grace and forgiveness. Or as the Catechism puts it:

"Therefore I tell you, every sin and blasphemy will be forgiven men, but the blasphemy against the Spirit will not be forgiven." There are no limits to the mercy of God, but anyone who deliberately refuses to accept his mercy by repenting, rejects the forgiveness of his sins and the salvation offered by the Holy Spirit. Such hardness of heart can lead to final impenitence and eternal loss.

The analogy I've heard is that it's to know God exists and know His power, but to believe you can do better. If you're familiar with the story of the fall of Lucifer/Satan, it's essentially his actions in a nutshell: he knew God exists and he knew God's divine grace, yet he openly rejected it.

And thats why it's so unforgivable: God can't forgive you or save you if you reject His ability to do so.