The Bible says that Moses was the meekest man on earth. Can we really believe that to be true considering all the things he did, such as the angry outbursts which eventually cost him the promised land and the grinding of the burnt golden calf?
The difficulty you are running into is in the definition of the word "meek." In our current usage, meekness is viewed as a negative trait. Current definitions include: "Humble in spirit or manner; suggesting retiring mildness or even cowed submissiveness," or "Compliant: evidencing little spirit or courage; overly submissive or compliant."
In discussing the Greek word for meekness, Vines stated, "The meaning of prautes is not readily expressed in English, for the terms meekness, mildness, commonly used, suggest weakness and pusillanimity to a greater or less extent, whereas prautes does nothing of the kind. Nevertheless, it is difficult to find a rendering less open to objection than 'meekness'; 'gentleness' has been suggested, but as prautes describes a condition of mind and heart, and as 'gentleness' is appropriate rather to actions, this word is no better than that used in both English Versions. ... Described negatively, meekness is the opposite to self-assertiveness and self-interest; it is equanimity of spirit that is neither elated nor cast down, simply because it is not occupied with self at all." It is that last phrase that I think captures the concept of meekness. It is the character trait where a person is unconcerned about himself. It is a total lack of self-pride. It is related to the concept of humility, but not quite the same thing.
In the Old Testament, the Hebrew word commonly translated as "meek" is anaw. It means someone who is afflicted. Understanding this lends greater meaning to the passage in which it was used. "Then Miriam and Aaron spoke against Moses because of the Ethiopian woman whom he had married; for he had married an Ethiopian woman. So they said, "Has the LORD indeed spoken only through Moses? Has He not spoken through us also?" And the LORD heard it. (Now the man Moses was very humble, more than all men who were on the face of the earth.)" (Numbers 12:1-3). Moses' brother and sister had been giving him grief because of the woman he chose to marry. They openly spoke rebelliously against Moses insinuating that since God had spoken through them as well as Moses, there was no need for them to follow Moses if they didn't want to do so. The statement about Moses' meekness (or humility in this translation) is an explanation of the circumstances. Though Moses suffered the scorn of his brother and sister, he did not take his problem to the Lord. He chose instead to bear the burden. Therefore, this explains why God chose to do something about the matter even though He wasn't asked. Moses wasn't the type of person to bring his personal problems to God.
Another example of Moses' meekness is found in Exodus 18:13-23 "And so it was, on the next day, that Moses sat to judge the people; and the people stood before Moses from morning until evening. So when Moses' father-in-law saw all that he did for the people, he said, "What is this thing that you are doing for the people? Why do you alone sit, and all the people stand before you from morning until evening?" And Moses said to his father-in-law, "Because the people come to me to inquire of God. When they have a difficulty, they come to me, and I judge between one and another; and I make known the statutes of God and His laws." So Moses' father-in-law said to him, "The thing that you do is not good. Both you and these people who are with you will surely wear yourselves out. For this thing is too much for you; you are not able to perform it by yourself. Listen now to my voice; I will give you counsel, and God will be with you: Stand before God for the people, so that you may bring the difficulties to God. And you shall teach them the statutes and the laws, and show them the way in which they must walk and the work they must do. Moreover you shall select from all the people able men, such as fear God, men of truth, hating covetousness; and place such over them to be rulers of thousands, rulers of hundreds, rulers of fifties, and rulers of tens. And let them judge the people at all times. Then it will be that every great matter they shall bring to you, but every small matter they themselves shall judge. So it will be easier for you, for they will bear the burden with you. If you do this thing, and God so commands you, then you will be able to endure, and all this people will also go to their place in peace."" Here was a man literally wearing himself out trying to help everyone solve their problems, while never complaining or thinking about what it was doing to him. Moses willing bore the difficulty of work given to him. It took his father-in-law to see the problem and tell Moses that he need to tell God about the problem and start training men to assume some of the burden he was bearing on behalf of the people.
Jesus described himself as meek, "Take My yoke upon you and learn from Me, for I am gentle and lowly in heart, and you will find rest for your souls" (Matthew 11:29). By this he is not saying was retiring or cowed, after all recall his driving out the moneychangers from the temple on two occasions (John 2:14-17 and Matthew 21:12-13). Rather he is inviting us to take up his light burden for us as he bears our heavy burdens on our behalf. "Surely He has borne our griefs And carried our sorrows; Yet we esteemed Him stricken, Smitten by God, and afflicted. But He was wounded for our transgressions, He was bruised for our iniquities; The chastisement for our peace was upon Him, And by His stripes we are healed. All we like sheep have gone astray; We have turned, every one, to his own way; And the LORD has laid on Him the iniquity of us all. He was oppressed and He was afflicted, Yet He opened not His mouth; He was led as a lamb to the slaughter, And as a sheep before its shearers is silent, So He opened not His mouth" (Isaiah 53:4-7). Jesus bore so much on our behalf and without a sound of complaint. That is meekness.
It is such a trait that every Christian is to cultivate. "Therefore, as the elect of God, holy and beloved, put on tender mercies, kindness, humility, meekness, longsuffering; bearing with one another, and forgiving one another, if anyone has a complaint against another; even as Christ forgave you, so you also must do" (Colossians 2:12-13). "Brethren, if a man is overtaken in any trespass, you who are spiritual restore such a one in a spirit of gentleness, considering yourself lest you also be tempted. Bear one another's burdens, and so fulfill the law of Christ" (Galatians 6:1-2).