A Synagogue Ruler
That a "synagogue ruler" should come to Jesus would be notable. Jesus wasn't particularly friendly with the religious establishment. He frequently annoyed them (see this parable  -- and the authorities'  reaction -- or this one , for example). But Jairus was sufficiently desperate -- his daughter was deathly ill -- that he sought Jesus out and begged him to come.

It is in the context of this request, and the resulting hurry back to Jairus' house, that this encounter of the sick -- and poor! -- woman takes place. This is significant for two reasons. First, Jesus took (precious!) time away from an important "client" -- religious, male, and rich -- to minister to an unimportant one -- sickly, female, and poor. But perhaps even more significant, Jesus' pause to address the sick woman made it clear to everyone that he had been touched by someone who was "unclean". The synagogue ruler would have grasped the significance of this: an "unclean" touch, according to the law, rendered Jesus also "unclean". And a "synagogue ruler" would not have permitted an "unclean" person to enter his house!

But Jesus' entrance into Jairus' house indicates that Jairus accepted that Jesus was no ordinary man. The customary understanding did not apply to him! 
Mark 5:21 (ESV)  And when Jesus had crossed again in the boat to the other side, a great crowd gathered about him, and he was beside the sea. 22 Then came one of the rulers of the synagogue, Jairus by name, and seeing him, he fell at his feet 23 and implored him earnestly, saying, “My little daughter is at the point of death. Come and lay your hands on her, so that she may be made well and live.”
(verses 24-34 represent another encounter -- full text for reference )
35 While he was still speaking, there came from the ruler's house some who said, “Your daughter is dead. Why trouble the Teacher any further?” 36 But overhearing what they said, Jesus said to the ruler of the synagogue, “Do not fear, only believe.” 37 And he allowed no one to follow him except Peter and James and John the brother of James. 38 They came to the house of the ruler of the synagogue, and Jesus saw a commotion, people weeping and wailing loudly. 39 And when he had entered, he said to them, “Why are you making a commotion and weeping? The child is not dead but sleeping.” 40 And they laughed at him. But he put them all outside and took the child's father and mother and those who were with him and went in where the child was. 41 Taking her by the hand he said to her, “Talitha cumi,” which means, “Little girl, I say to you, arise.” 42 And immediately the girl got up and began walking (for she was twelve years of age), and they were immediately overcome with amazement. 43 And he strictly charged them that no one should know this, and told them to give her something to eat.

Jesus' touch of the "unclean" resulted in her cleansing -- without defiling Jesus. Having accepted this, Jairus could then experience a related truth: Jesus' touch of the dead (also considered cause for defilement in the law) could result in  life