I think it is really important that we look at these verses in context and compare them with other verses in the New Testament on this issue. Here is the context.
Mat 7:1-5
(1) Judge not, that ye be not judged.
(2) For with what judgment ye judge, ye shall be judged: and with what measure ye mete, it shall be measured to you again.
(3) And why beholdest thou the mote that is in thy brother's eye, but considerest not the beam that is in thine own eye?
(4) Or how wilt thou say to thy brother, Let me pull out the mote out of thine eye; and, behold, a beam is in thine own eye?
(5) Thou hypocrite, first cast out the beam out of thine own eye; and then shalt thou see clearly to cast out the mote out of thy brother's eye.
The point being made here is not that we should not judge. That interpretation is to lift the first verse out of context entirely. The point is that we will be judged with the same judgement that we have toward others. Before judging, we should be careful to allow God to deal with and cleanse us in the arena in which we are about to judge another. For example, If I have repented of bitterness and unforgiveness and walk in love, forgiveness, and grace, then I am able to judge properly another who is walking in bitterness or unforgiveness. But judgement in this sense does not mean condemnation. It means to differentiate between two things. It means to be able to identify and approach a fault or sin in another with the love of Christ and in the Spirit so that my judgement is just, righteous, and bears peaceable fruit.
Jesus dealt with just this scenario when He was rebuked for healing on the Sabbath. Joh 7:23-24
(23) If a man on the sabbath day receive circumcision, that the law of Moses should not be broken; are ye angry at me, because I have made a man every whit whole on the sabbath day?
(24) Judge not according to the appearance, but judge righteous judgment.
We are told that we will judge the world in the end, so we ought to be able, in the church, to judge right and wrong in matters of this world. 1Co 6:1-3
(1) Dare any of you, having a matter against another, go to law before the unjust, and not before the saints?
(2) Do ye not know that the saints shall judge the world? and if the world shall be judged by you, are ye unworthy to judge the smallest matters?
(3) Know ye not that we shall judge angels? how much more things that pertain to this life?
Again, it is not condemnation in the sense of passing final judgement in this world, but it is a distinguishing between right and wrong. Paul goes on to advise that we ought to set those least esteemed in the body over this work of judging who is right and who is wrong in a dispute between brothers.
The issue with judgment is that it comes back to us in the same form it goes out from us. If I deal with my brothers and sisters in the Spirit, acting in the love of God and humility, that is the judgement that will exercised toward me. If I judge harshly, proudly, and in my flesh, that is what I can expect to be judged by in my failures as well.