The teachings of Jesus can be summarized in a single word - love.
Jesus primarily taught us to do two things -- love our God, and love one another. In fact, the teachings of the entire Bible serve the same purpose. Throughout the Bible, we are shown the wonders of God. We are shown that God is powerful, and wrathful, and faithful. We are introduced to this wonderful God, shown what he has done for us, and are told that we should love him. We are also shown how to love him and praise him. When God came to dwell among us, he taught this same basic principle. By loving God, we obey his commandments and live to please him, much like we would an earthly father who treated us well and loved us thoroughly.
Jesus also taught us to love one another. This is directly related to loving God, since it pains any father to see his children not love one another. When a man kills another, God sees it as one of his children killing another. For obvious reasons, this does not please God one bit. Each and every one of the Ten Commandments tells us to love God and each other:
Each of these laws tell us to love either God or each other. The first four tell us to love and respect our maker, the rest tell us to love each other. At least in the spiritual sense, we must think of every person on this earth as being our brother or sister. With this in mind, one can easily see why God gave us these commandments. We wouldn't kill our brother, or steal from our brother, or lie to get our brother in trouble. We're also told to respect our parents, and to respect others by not committing adultery, or stealing, or lying to get others in trouble. The last commandment summarizes the four before it, because coveting your neighbor's wife or goods could easily lead to adultery, killing, or stealing.
The teachings of Jesus reinforced these commandments. Jesus emphasized a strong love for God and also taught us to treat others well. In fact, some have referred to this as the "Golden Rule", which states "Do unto others as you would have them do unto you." In other words, treat other people however you would want them to treat you.
To give you an idea of the teachings of Jesus, here is an excerpt from the Bible in which Jesus speaks of the Ten Commandments: (Click here for larger text)
KJV Matthew 5:21-48
21 Ye have heard that it was said by them of old time, Thou shalt not kill; and whosoever shall kill shall be in danger of the judgment:
22 But I say unto you, That whosoever is angry with his brother without a cause shall be in danger of the judgment: and whosoever shall say to his brother, Raca, shall be in danger of the council: but whosoever shall say, Thou fool, shall be in danger of hell fire.
23 Therefore if thou bring thy gift to the altar, and there rememberest that thy brother hath ought against thee;
24 Leave there thy gift before the altar, and go thy way; first be reconciled to thy brother, and then come and offer thy gift.
25 Agree with thine adversary quickly, whiles thou art in the way with him; lest at any time the adversary deliver thee to the judge, and the judge deliver thee to the officer, and thou be cast into prison.
26 Verily I say unto thee, Thou shalt by no means come out thence, till thou hast paid the uttermost farthing.
27 Ye have heard that it was said by them of old time, Thou shalt not commit adultery:
28 But I say unto you, That whosoever looketh on a woman to lust after her hath committed adultery with her already in his heart.
29 And if thy right eye offend thee, pluck it out, and cast it from thee: for it is profitable for thee that one of thy members should perish, and not that thy whole body should be cast into hell.
30 And if thy right hand offend thee, cut it off, and cast it from thee: for it is profitable for thee that one of thy members should perish, and not that thy whole body should be cast into hell.
31 It hath been said, Whosoever shall put away his wife, let him give her a writing of divorcement:
32 But I say unto you, That whosoever shall put away his wife, saving for the cause of fornication, causeth her to commit adultery: and whosoever shall marry her that is divorced committeth adultery.
33 Again, ye have heard that it hath been said by them of old time, Thou shalt not forswear thyself, but shalt perform unto the Lord thine oaths:
34 But I say unto you, Swear not at all; neither by heaven; for it is God's throne:
35 Nor by the earth; for it is his footstool: neither by Jerusalem; for it is the city of the great King.
36 Neither shalt thou swear by thy head, because thou canst not make one hair white or black.
37 But let your communication be, Yea, yea; Nay, nay: for whatsoever is more than these cometh of evil.
38 Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth:
39 But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.
40 And if any man will sue thee at the law, and take away thy coat, let him have thy cloak also.
41 And whosoever shall compel thee to go a mile, go with him twain.
42 Give to him that asketh thee, and from him that would borrow of thee turn not thou away.
43 Ye have heard that it hath been said, Thou shalt love thy neighbour, and hate thine enemy.
44 But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you;
45 That ye may be the children of your Father which is in heaven: for he maketh his sun to rise on the evil and on the good, and sendeth rain on the just and on the unjust.
46 For if ye love them which love you, what reward have ye? do not even the publicans the same?
47 And if ye salute your brethren only, what do ye more than others? do not even the publicans so?
48 Be ye therefore perfect, even as your Father which is in heaven is perfect.
In these verses, Jesus is discussing common ideas and laws which the Jews often consider the foundation of their society, among them the Ten Commandments and Proverbs. Most of what Jesus discusses above comes from Proverbs, the book of the Bible which contains the written wisdom of King Solomon, third king of Israel. Solomon was reknowned for his wisdom and is largely responsible for many of the foundational laws that society holds dear today, especially in the Middle East. Even so, Solomon's laws were only expanding on the original Ten Commandments. Most of Solomon's laws were very black and white. If this crime is committed, this punishment shall be given. Plain and simple. But full of holes. There are always extenuating circumstances. Most issues in life are gray, not black and white. What Jesus is doing here is filling in the holes and gaps. In every case, the core theme is love. Although there are times when peace isn't an option, these teachings show us that we should show mercy and generosity whenever possible. Had we been left to fill in the holes ourselves, it's highly unlikely we would be so merciful.
Many have used Jesus' teachings against him, stating that he is contradicting the law, and therefore the Word of God. In fact, this was the chief way in which the Jewish religious leaders attempted to discredit Jesus in the Bible. However, a closer look reveals that Jesus never once contradicts. He only expands upon the original text. Of course, at first glance, it will seem like contradiction at places. Many people choose to stop looking after that initial glance. If you continue to look, though, it always fits together somehow.
Of course, the teachings of Jesus in their entirety are far too numerous to be contained here. To learn more of Jesus and what he taught, pick up a Bible and read through the books of the New Testament. It's all in there.