A Sick Woman
The woman in this story is clearly in bad shape, having suffered with an uncomfortable condition for twelve years.  She must have heard what Jesus could do, and been curious whether any of it was true.

But in order to understand this encounter, it helps to have a bit of background information. The law of Moses, which was given for the benefit of the community as a whole, wasn't always compassionate to individuals on the fringes. This woman's condition would have made her "unclean", keeping her out of polite society (see Leviticus 15:25).

The fact that the crowd around Jesus is led by a "synagogue leader" and his retinue is also important: in order to gain access to Jesus, this woman would by unable to avoid touching people (thus rendering them "unclean"). Respectable religious people took that kind of thing very seriously.

But the sick woman is sufficiently desperate: she is willing to risk the wrath of the community for the chance to touch Jesus. Clearly, her plan is to do it as subtly as possible. But Jesus foils her plan: he is aware of her touch, and asks who it was that touched him.
Mark 5:21 (ESV) When Jesus had again crossed over by boat to the other side of the lake, a large crowd gathered around him while he was by the lake. 22 Then one of the synagogue leaders, named Jairus, came, and when he saw Jesus, he fell at his feet. 23 He pleaded earnestly with him, “My little daughter is dying. Please come and put your hands on her so that she will be healed and live.” 24 So Jesus went with him.
A large crowd followed and pressed around him.
 
25 And a woman was there who had been subject to bleeding for twelve years. 26 She had suffered a great deal under the care of many doctors and had spent all she had, yet instead of getting better she grew worse. 27 When she heard about Jesus, she came up behind him in the crowd and touched his cloak, 28 because she thought, “If I just touch his clothes, I will be healed.” 29 Immediately her bleeding stopped and she felt in her body that she was freed from her suffering.
30 At once Jesus realized that power had gone out from him. He turned around in the crowd and asked, “Who touched my clothes?”
31 “You see the people crowding against you,” his disciples answered, “and yet you can ask, ‘Who touched me?’ ”
32 But Jesus kept looking around to see who had done it. 33 Then the woman, knowing what had happened to her, came and fell at his feet and, trembling with fear, told him the whole truth. 34 He said to her, “Daughter, your faith has made you whole. Go in peace and be freed from your suffering.” ( within the context of this encounter )

She is clearly horrified by his discovery, but he quickly comforts her, wanting only her wholeness, her peace, and her freedom. Instead of Jesus' being defiled by her touch (in keeping with common perception), she was healed by his.