Jesus did exist
Not only do we have four records of his life and work called the Gospels but there is evidence from non-Christian sources that he existed. In AD 52 a man called Thalles wrote an account of the Mediterranean world mentioning Jesus being crucified, Pliny a governor in a Roman province wrote to the Emperor in AD 112 and mentioned Jesus and his followers. Tacitus, a Roman historian writing in AD 115 speaking of Christians referred to a Christus from whom their name is derived.
The most striking reference is a Jewish historian called Josephus writing towards the end of the first century said Now, there was about this time Jesus ,a wise man, if it be lawful to call him a man, for he was a doer of wonderful works...
None of these people were sympathetic towards Jesus or Christianity.
No-one taught like Jesus
He spoke as if he had the direct authority of God himself. The religious teachers backed up their sayings by quoting earlier religious leaders. Jesus didnt do this, he prefixed his sayings with the term Amen - this is so.
Here are a few samples:
"You have heard that it was said, `Eye for eye, and tooth for tooth.' But I tell you, Do not resist an evil person. If someone strikes you on the right cheek, turn to him the other also. And if someone wants to sue you and take your tunic, let him have your cloak as well. If someone forces you to go one mile, go with him two miles. Give to the one who asks you, and do not turn away from the one who wants to borrow from you.
"You have heard that it was said, `Love your neighbour and hate your enemy.' But I tell you: Love your enemies and pray for those who persecute you, that you may be sons of your Father in heaven. He causes his sun to rise on the evil and the good, and sends rain on the righteous and the unrighteous. If you love those who love you, what reward will you get? Are not even the tax collectors doing that? And if you greet only your brothers, what are you doing more than others? Do not even pagans do that? Be perfect, therefore, as your heavenly Father is perfect.
"Not everyone who says to me, `Lord, Lord,' will enter the kingdom of heaven, but only he who does the will of my Father who is in heaven.
(Note - "Lord" is a term reserved for God alone and yet here is Jesus using it of himself and speaking of my Father in heaven - not our Father - no first century Jew would dare to do that)
Jesus puts his own words on a par with Gods so that peoples response to him determines their eternal destiny:
"Therefore everyone who hears these words of mine and puts them into practice is like a wise man who built his house on the rock. The rain came down, the streams rose, and the winds blew and beat against that house; yet it did not fall, because it had its foundation on the rock. But everyone who hears these words of mine and does not put them into practice is like a foolish man who built his house on sand. The rain came down, the streams rose, and the winds blew and beat against that house, and it fell with a great crash."
The peoples reaction?
When Jesus had finished saying these things, the crowds were amazed at his teaching, because he taught as one who had authority, and not as their teachers of the law.
No one lived like Jesus
His life was remarkable. Even his enemies couldnt find fault and had to make up charges against him (Matthew 26:59). Not only did he welcome the outcasts of society into his company he backed up what he said by what he did. To claim the authority of God is one thing - to act it out is another.
Who can forgive sins but God? - yet Jesus did (Mark 2:1-12) In the Old Testament God chose the 12 tribes of Israel as the nucleus of his people, Jesus chooses 12 disciples deliberately mirroring this (Mark 3:13). For the Jews only God could control nature (Psalm 89:9).We see Jesus doing this (Mark 4:35). Only God is to be worshipped, yet Jesus accepted such worship (John 20:28).
No one died like Jesus
Of course thousands died by crucifixion. But Jesus claimed his death was both something he was willing to do and part of a divine plan, dying as he did at such an early age (33):
"We are going up to Jerusalem," he said, "and the Son of Man will be betrayed to the chief priests and teachers of the law. They will condemn him to death and will hand him over to the Gentiles, who will mock him and spit on him, flog him and kill him. Three days later he will rise."
For even the Son of Man did not come to be served, but to serve, and to give his life as a ransom for many.
There was a divine must about it fulfilling the prophecies of Scripture (eg Psalm 22 and Isaiah 52-53)
It was part of Gods rescue mission to take away the guilt of our sin by dying as a sacrifice in our place - The judge being judged in our place.
No one was raised from the dead like Jesus.
This is the subject of another question - Did Jesus really rise from the dead? . But whereas there is evidence of people being resuscitated, even by Jesus, in that they died and were brought back to life, Jesus' resurrection was different in that his body had been changed never to die again. Evidence that his death achieved its purpose as a sacrifice for sin.
As the writer CS Lewis pointed out - Jesus was either mad (claiming divinity) or bad (leading people astray) or God -the divine Creator as a perfect human being. We have to choose.