Jesus is in worship on the Sabbath and is confronted with a man who needs healing. The man has a withered hand, this is not life threatening nor, it could be argued, was it so critical that it could not wait until the next day. The teachers of the Law were very clear in their understanding of the commandment to not do ANY work on the Sabbath. In order to help the people from breaking the commandment the teachers had developed detailed definitions of what constituted work. Jesus problem with the teachers is that by their definition if you are healer then healing constituted work for you and therefore was forbidden on the sabbath. Jesus response, elsewhere, that "sabbath was made for humans not humans for the sabbath" is his way of saying that the overlay of religious rules over a basic commandment is more destructive than freeing and misses the purpose of the commandment.
Mark 3 contains one of the most difficult passages in the Gospel: "blasphemy against the Holy Spirit". The scribes had been saying that Jesus was doing what he was doing (healing, casting out demons, etc.) because he (Jesus) had an evil spirit -- that he was in league with the devil. Calling good evil (in this case calling the Holy Spirit an Evil Spirit or unclean spirit -- aka demon) is negating the possibility of ever leaving evil and returning to good. This would deny the possibility of redemption because this confusion would keep people from turning to the one source of salvation. This particular act could not be done accidentally or casually but would constitute a complete rejection of the plan and person God sent to redeem the world. In short, blasphemy against the Holy Spirit would be to refuse the salvation God has provided for us through Jesus -- you cannot be forgiven unless you choose to accept forgiveness.
The end of Mark 3 gives us reference to Jesus Mother and his brothers. Jesus brothers are listed by name in Matthew 13:55-56. James (author of the Epistle of James) is called "the Lord's half brother" as is Jude (the author of Jude). Jesus family plays a significant role in the early days of Christianity in and around Jerusalem.