The Gospel of Mark is the only book in the Bible that records Jesus sighing.  The two accounts are in back to back chapters - Mark 7:34 and Mark 8:12.

In the first instance, Jesus encounters a deaf and mute man.  Here is the first event.

31 Then Jesus left the vicinity of Tyre and went through Sidon, down to the Sea of Galilee and into the region of the Decapolis. 32 There some people brought to him a man who was deaf and could hardly talk, and they begged him to place his hand on the man. 33 After he took him aside, away from the crowd, Jesus put his fingers into the man's ears. Then he spit and touched the man's tongue. 34 He looked up to heaven and with a deep sigh said to him, ""Ephphatha!"" (which means, "Be opened!"). 35 At this, the man's ears were opened, his tongue was loosened and he began to speak plainly." 

Notice that this was not a healing requiring the sick to display a level of faith.  This was all on Jesus!  Jesus healed this man by taking him away from the crowds, placing His fingers in the man's ears (o.k.) and as Jesus' fingers are in the man's ears He spit somewhere.  Then Jesus touched the man's tongue (with the same fingers that had been in those ears - yuck Jesus!).  Hey!  If I am that man and this is the formula that is going to open my ears and loosen my tongue then I'm saying, "So be it!".    Then we get to Jesus looking up to heaven "and with a deep sigh said to him, 'Ephphatha!' (which means, 'Be Opened!)."

When I asked the Holy Spirit why did Jesus sigh here?  The Holy Spirit told me, "Examine Matthew 15:29-32 with verse 32 as the key."  Jesus had just healed many people from many afflictions and now He observes another need.  The masses of people before Him are hungry.  The verse reads, "Jesus called his disciples to him and said, 'I have compassion for these people; they have already been with me three days and have nothing to eat. I do not want to send them away hungry, or they may collapse on the way.'"  

The Holy Spirit highlighted "I have compassion for these people;" out of this text and elaborated - "This is the reason I just healed them and this is the reason I am going to feed them - "I have compassion for these people." At this point I sighed deeply too and said, "Thank you Lord that your compassions never fail (Lamentations 3:22)."  

The surroundings, atmosphere, and situation were very different for the second occasion in which Jesus sighed.  It is just after Jesus miraculously fed the four thousand and He and the His Disciples are leaving -

"The people ate and were satisfied. Afterward the disciples picked up seven basketfuls of broken pieces that were left over.  About four thousand men were present. And having sent them away, he got into the boat with his disciples and went to the region of Dalmanutha.  The Pharisees came and began to question Jesus. To test him, they asked him for a sign from heaven.  He sighed deeply and said, "Why does this generation ask for a miraculous sign? I tell you the truth, no sign will be given to it."  Then he left them, got back into the boat and crossed to the other side."  (Mark 8:8-13)    

The Pharisees were a bothersome group of know-it-alls that desired to display their knowledge to the world and if anyone disagreed with them the consequences would be dire.  They followed Jesus around like a bunch of buzzards or vultures wanting to pick at the meat of an already dead animal or man.  

I translate this sigh as a sigh of frustration.  I am not sure what kind of sign the Pharisees were looking for. Jesus had provided for the people.  The Scriptures prophesied to His coming and work.  The Pharisees supposedly knew all this.  But still the Pharisees ask Jesus for a sign from heaven.  Jesus sighs deeply, questions their request, passes judgment ("no sign will be given to it") [Matthew 12:39 records the Sign of Jonah will be given], then left the company of the Pharisees, got back in the boat and went to the complete other side of the lake.  I don't think Jesus cared to much for the Pharisees ways.  He died for them too though.  What a Savior!

From just these two instances of Jesus sighing we find that "we do not have a high priest who is unable to sympathize with our weaknesses, but we have one who has been tempted in every way, just as we are--yet was without sin." (Hebrews 4:15)  Jesus felt our pain and had compassion on the people by releasing them from their variety of "prisons".  Jesus also knew and knows our frustrations with anything and everything, but in this case, especially the know-it-alls of the world.  When Jesus sighed He understood and experienced the humanity of us all, but did not give into any sinful emotions that may have followed.  So, anytime you hear a sigh or you have to sigh yourself, remember the meaning that followed - When Jesus Sighed.