Matthew 7:1-5 is likely one of the most mis-used and abused passages in all of the New Testament. How often do we hear, both from not only those who have rejected Christ's gift of Eternal Life, but even from those who have professed Christ as their Lord and Savior just the first verse "Judge not, that ye be not judged"... or even more selectively out of context "Judge Not!". Further, how often do we hear this put forward as some kind of "proof" that Christians are not suppose to pass any form of judgment on anyone else, under any circumstances?
One cannot take verse 1 alone - but instead must take it in context of at least the next 4 verses - and also look at WHO Jesus was addressing - The Pharisees, who were judging everyone by their own standards (that had grown to be completely outside of God's commands in His Word). Instead, they had substituted some adulterated and completely unscriptural "Code of Conduct" for God's otherwise relatively simple Law. They were fond of pointing fingers.
The beam in their own eyes was their blindness to God's Word. It was their own man-created codes. Even worse, they often didn't live up to the very standard they held everyone else up to.
What is clear here is that Jesus was addressing hypocrites who wanted the world to live by their misguided standards. What the Pharisees needed was to remove these "beams" that were blinding them to reality.
When we come to Christ, we too remove the "beam" from our eyes - making us better equipped to help others see the "mete" - aka - splinter or sin in their own lives.
This is one of the many purposes of assembling together in a local New Testament church - to hold each other up, to help each other resist temptation, to preserve our testimony, and to live lives that are, as Paul put it, "I beseech you therefore, brethren, by the mercies of God, that ye present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service." (Romans 12:1)
Judgment by man-invented standards, would thus be wrong. Judgement by God's standards is NOT wrong. But the spirit of such "judgment" should be a spirit of love and restoration.
Jesus told those hearing him (Luke 12:54-59) to judge what He said - according to the scriptures. We are not to judge according to personal preferences, but according to the Word of God.
Christ specifically endorsed judgment by men - but not based on appearance, but on what a person does and/or continues to do as compared to "righteousness" - moral issues for which the standard is clearly set in God's Word.
Hand-in-hand with this (indeed, probably best exercised before) would be the admonition to examine ourselves regularly (2 Corinthians 13:5-7)