Matthew chapter 6 talks about the left hand not knowing what the right hand is doing. What does this mean and biblically what is the meaning of left and right hand in the Bible?

Let's begin by looking at the passage:

Matthew 6:1-4 [1] Take heed that ye do not your alms before men, to be seen of them: otherwise ye have no reward of your Father which is in heaven. [2] Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men. Verily I say unto you, They have their reward. [3] But when thou doest alms, let not thy left hand know what thy right hand doeth: [4] That thine alms may be in secret: and thy Father which seeth in secret himself shall reward thee openly.
See All... Take heed that ye do not your alms before men, to be seen of them: otherwise ye have no reward of your Father which is in heaven. Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men. Verily I say unto you, They have their reward. But when thou doest alms, let not thy left hand know what thy right hand doeth: That thine alms may be in secret: and thy Father which seeth in secret himself shall reward thee openly.

Matthew, chapters 5-7, contains what is known as the Sermon on the Mount. Matthew 6:1-21 [1] Take heed that ye do not your alms before men, to be seen of them: otherwise ye have no reward of your Father which is in heaven. [2] Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men. Verily I say unto you, They have their reward. [3] But when thou doest alms, let not thy left hand know what thy right hand doeth: [4] That thine alms may be in secret: and thy Father which seeth in secret himself shall reward thee openly. [5] And when thou prayest, thou shalt not be as the hypocrites are: for they love to pray standing in the synagogues and in the corners of the streets, that they may be seen of men. Verily I say unto you, They have their reward. [6] But thou, when thou prayest, enter into thy closet, and when thou hast shut thy door, pray to thy Father which is in secret; and thy Father which seeth in secret shall reward thee openly. [7] But when ye pray, use not vain repetitions, as the heathen do: for they think that they shall be heard for their much speaking. [8] Be not ye therefore like unto them: for your Father knoweth what things ye have need of, before ye ask him. [9] After this manner therefore pray ye: Our Father which art in heaven, Hallowed be thy name. [10] Thy kingdom come, Thy will be done in earth, as it is in heaven. [11] Give us this day our daily bread. [12] And forgive us our debts, as we forgive our debtors. [13] And lead us not into temptation, but deliver us from evil: For thine is the kingdom, and the power, and the glory, for ever. Amen. [14] For if ye forgive men their trespasses, your heavenly Father will also forgive you: [15] But if ye forgive not men their trespasses, neither will your Father forgive your trespasses. [16] Moreover when ye fast, be not, as the hypocrites, of a sad countenance: for they disfigure their faces, that they may appear unto men to fast. Verily I say unto you, They have their reward. [17] But thou, when thou fastest, anoint thine head, and wash thy face; [18] That thou appear not unto men to fast, but unto thy Father which is in secret: and thy Father, which seeth in secret, shall reward thee openly. [19] Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt, and where thieves break through and steal: [20] But lay up for yourselves treasures in heaven, where neither moth nor rust doth corrupt, and where thieves do not break through nor steal: [21] For where your treasure is, there will your heart be also.
See All... deals with the importance of laying up treasures in heaven instead of piling them up on earth. One of the major ways we accomplish this is by doing our good deeds quietly and not as a show to be seen by others. Jesus gives three examples of this: the giving of alms (v.1-4), prayer (v.5-15), and fasting (v.16-18). Those who do these good deeds in a showy way in order to be seen of men have already received their reward--the praise of men. However, they will receive no additional heavenly reward for such actions. However, those who do these things secretly (that is, without a big show) have laid up treasures in heaven (v.19-21).

The giving of alms is the giving of money to help the poor and needy. No one should give alms to be seen of men (v.1); they should not sound a trumpet announcing their good deed (v.2); but should do the deed in secret in order to receive God's reward (v.4). Verse three and its teaching about not allowing the left hand to know what the right hand is doing is part of this argument.

When we do something good, we naturally feel good about ourselves. We tend to want to tell someone else so they can tell us how good we are. Jesus tells us that we should not be telling others of our good deeds. But if we keep rehearsing in our minds what a good thing we have done, we will tend to slip and tell others of the deed. The best way to keep from "spilling the beans" is to avoid thinking about it ourselves; to avoid making our actions a part of our inner conversation. This, I believe, is what Jesus is speaking of here.

Our right hand and our left hand are both a part of us. When one of them lets the other know what it is doing, it is carrying on an inner conversation; a conversation in the brain. Jesus is telling us: Don't even think about it; don't congratulate yourself for your good deed; don't think about how few people would have done such a thing; don't dwell on the act. Even self-praise can lessen your reward. And, the more you think about it, the more likely you are to tell someone else. Don't even let your left hand know what your right hand is doing.