What Is The Unforgivable Sin?

Scripture tells us there IS a sin which cannot be forgiven:

31 Therefore I say to you: Every sin and blasphemy shall be forgiven men, but the blasphemy of the Spirit shall not be forgiven.32 And whosoever shall speak a word against the Son of man, it shall be forgiven him: but he that shall speak against the Holy Ghost, it shall not be forgiven him, neither in this world, nor in the world to come.  (Matthew 12:31-32)

28 Amen I say to you, that all sins shall be forgiven unto the sons of men, and the blasphemies wherewith they shall blaspheme:29 But he that shall blaspheme against the Holy Ghost, shall never have forgiveness, but shall be guilty of an everlasting sin. (Mark 3:28-29)

The answer would be "final impenitence," that is - an obstinate refusal to seek forgiveness of mortal sin which one knows and recognizes as mortal sin.  This is blaspheme for it is a direct denial of the healing power of the Holy Ghost.  To knowingly not seek such reconciliation is a sin against the Holy Ghost and to do so with such finality that one does not seek this up to and through death makes it unforgivable.  

The Catechism of the Catholic Church explains it this way:

CCC 1864 "Whoever blasphemes against the Holy Spirit never has forgiveness, but is guilty of an eternal sin." 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.

St. Thomas Aquinas offers several arguments in the Summa Theologica, but it seems summed up in the conclusion the CCC (above) makes:

According to the various interpretations of the sin against the Holy Ghost, there are various ways in which it may be said that it cannot be forgiven. For if by the sin against the Holy Ghost we understand final impenitence, it is said to be unpardonable, since in no way is it pardoned: because the mortal sin wherein a man perseveres until death will not be forgiven in the life to come, since it was not remitted by repentance in this life. 

Final impenitence - the refusal to seek forgiveness of ones sins.

St. Augustine says:  

"(Enchiridion lxxxiii) that 'he who dies in a state of obstinacy is guilty of the sin against the Holy Ghost,' and (De Verb. Dom., Serm. lxxi) that 'impenitence is a sin against the Holy Ghost,' and (De Serm. Dom. in Monte xxii), that 'to resist fraternal goodness with the brands of envy is to sin against the Holy Ghost,' and in his book De unico Baptismo (De Bap. contra Donat. vi, 35) he says that 'a man who spurns the truth, is either envious of his brethren to whom the truth is revealed, or ungrateful to God, by Whose inspiration the Church is taught,' and therefore, seemingly, sins against the Holy Ghost" (ST 2b:14:2, Sed Contra)."  (Qtd. on:  http://www.ewtn.com/library/ANSWERS/UNFORGIV.HTM - an article by James Akin).

Again, final impenitence is concluded to be the sin against the Holy Ghost for which there is no forgiveness.  It is a final and unforgivable sin because to not seek reconciliation for mortal sin in this life means it is not forgiven in this life - and if not forgiven in this life, it will not be forgiven in the next.  I could get longer winded about this - but I believe the answer is summarized above.  The cited articles above go into longer arguments and defenses if one is interested in further reading - and I would encourage further reading if one still has questions.