To Heal on the Sabbath

For background, see the following videos in these times: 

• 11:59–17:59 in John 5 (Part 1) 1–18 • The Heart of Legalism.
• 25:21–39:37 in John 9 • “I was blind, now I see.”.

Jesus did several healings on the Sabbath, see Miracles of Jesus. The scribes and Pharisees asked Jesus if it was permissible to heal a person on the Sabbath
(Mat 12:9–10, Mark 3:1–2, Luke 6:6–7). Jesus answered them: 

• "I ask you, is it lawful on the Sabbath to do good or to do harm, to save life or to destroy it?" (Luke 6:9, Mark 3:4).
• "Which one of you who has a sheep, if it falls into a pit on the Sabbath, will not take hold of it and lift it out? Of how much more value is a man than a sheep! So it is lawful to do good on the Sabbath." (Mat 12:11–12).

Jesus also said on another occasion:

•  "Which of you, having a son (a donkey in some manuscripts) or an ox that has fallen into a well on a Sabbath day, will not immediately pull him out?" (Luke 14:5).

And Jesus said on another occasion:

• "You hypocrites! Does not each of you on the Sabbath untie his ox or his donkey from the manger and lead it away to water it? And ought not this woman, a daughter of Abraham whom Satan bound for eighteen years,
  be loosed from this bond on the Sabbath day?" (Luke 13:15-16).

Jesus said on another occasion:

• "And if you had known what this means: 'I desire mercy, and not sacrifice' (Hosea 6:6 (LXX)), you would not have condemned the guiltless. For the Son of Man is lord of the Sabbath."
  (Mat 12:7–8, Luke 6:5).
• "The Sabbath was made for man, not man for the Sabbath.
  So the Son of Man is lord even of the Sabbath." (Mark 2:27–28).

And Jesus said on another occasion:

• "Go and learn what this means: 'I desire mercy, and not sacrifice' (Hosea 6:6 (LXX)). For I came not to call the righteous, but sinners."
  (Mat 9:13). 

What Jesus means is that mercy is more important than sacrifice. Mercy triumphs over following rules and regulations, see the end of The Book of Ruth. 
See also what the 4th commandment in The Ten Commandments means to us.

Jesus had a close relationship to God the Father, and did the will of the Father 
all the time (John 5:20, 6:38, 8:28–29, 12:49–50, 14:31a). See it explained in 
The Order between Jesus and the Father in the Trinity. This is exemplified when Jesus was persecuted for healing people on the Sabbath. Jesus then said: 

• "My Father is working until now, and I am working." (John 5:17).
• "Truly, truly, I say to you, the Son can do nothing of his own accord, but only what he sees the Father doing. For whatever the Father does,
  that the Son does likewise." (John 5:19).

Jesus did no work whatsoever when he healed them. Yet these religious leaders hated him for doing this on the Sabbath, and even wanted to kill Jesus for doing these miracles (Mat 12:14, Mark 3:6, Luke 6:11, John 5:18). It's very possible for religious people to get the perspective grossly wrong. They probably thought they were doing the right thing by wanting to destroy Jesus for healing people on the Sabbath. But it was a great violation of the Sabbath by these religious men when they planned to murder a godly man who had never sinned against anyone.

Jesus fulfilled the Law (Mat 5:17–20), but was confronted with empty religion and man-made rules, see Confronting Meaningless Religion. Jesus kept the Law perfectly (including keeping the Sabbath) as a man but also as the Son of God (Heb 4:15).
Keeping them perfect gave Jesus the ability to pay the debt we owe to the Law. Through his physical death on the cross, he payed the price and fulfilled the requirements of the Law for us all (Heb 9:15), see The Crucifixion of Jesus. Whoever who is hung on a pole is under God’s curse (Gal 3:13, Deu (5 Mos) 21:23). Jesus took the curse for us. By doing so, he satisfied the requirements of the Law. Our part is to admit it, believe and trust Jesus, and invite him into our heart
(John 1:12), see Salvation. See this further explained in Jesus and the Law.

See also Some stories in the gospels at the end of The Four Gospels.