1 John Chapter 1 & 3 - Is there an apparent contradiction? I have often wondered why the author of 1 John, in the beginning of his letter, seems to claim that even though people have accepted Christ they will continue to sin, but two chapters later he seems to be making the opposite claim? I've never found an adequate explanation to this. Why is this?

1 John 1:8, 10: "If we claim to be without sin, we deceive ourselves and the truth is not in us. If we confess our sins, he is faithful and just and will forgive us our sins and purify us from all unrighteousness. If we claim we have not sinned, we make him out to be a liar and his word has no place in our lives."

1 John 3:9: "No one who is born of God will continue to sin, because God's seed remains in him; he cannot go on sinning, because he has been born of God."
In a case like this, it's best to go back to the original Greek.  As jrista pointed out, the key verb in 3:9 is the one your version translates as "continue to", in the greek "ποιέω".  This Greek Lexicon gives a lot of translations for this verb, including:


  to make; with the names of things made, to produce, construct, form, fashion, etc.; to be the authors of, the cause


This same verb is used in the immediately preceding verses, 3:7-8, which the NIV translates:


  Dear children, do not let anyone lead you astray. He who does what is
  right is righteous, just as he is righteous. He who does what is
  sinful is of the devil, because the devil has been sinning from the
  beginning.


The emphasis here seems to be on what you produce, cause or make.  As such, I would interpret it similarly to Jesus' statement in Matthew 7:18,20:


  A good tree cannot bear bad fruit, and a bad tree cannot bear good
  fruit...Thus, by their fruit you will recognize them.


The point of both passages does not seem to be saying that any who are "born of God" will never sin, but that the product of their lives will be to "produce fruit in keeping with repentance." (Matthew 3:8)  They may sin, but they are saved by genuine repentance such that their lives are characterized by righteousness and good fruit, not sin.