The interpretation is straightforward: He who rebukes others should take care that he himself is blameless. Theophylact (11th cent. Byzantine) summarizes as follows:

Judge not, that ye be not judged.

He forbids condemning others. A reproof is for another's benefit, but condemnation expresses only derision and scorn. You may also understand that the Lord is speaking of one who, despite his own great sins, condemns others who have lesser sins of which God will be the judge.

For with what judgment ye judge, ye shall be judged: and with what measure ye mete, it shall be measured to you again. And why beholdest thou the speck [KJV: mote] that is in thy brother’s eye, but considerest not the beam that is in thine own eye? Or how wilt thou say to thy brother, Let me pull out the speck out of thine eye; and, behold, a beam is in thine own eye? Thou hypocrite, first cast out the beam out of thine own eye; and then shalt thou see clearly to cast out the speck out of thy brother’s eye.

He who would rebuke others ought to be blameless himself. If he himself has a plank in his eye, that is, some great sin, and he finds fault with another who has only a speck, he causes that man to be even more shameless in his sin. The Lord shows that he who has sinned greatly is not even able to see clearly the sin of his brother. For how could one who has a plank in his eye even see another man who is only slightly injured.

The Explanation of the Holy Gospel According to Saint Matthew

Although these are good answers about how God feels about judging, I think that there is a much more simpler truth to all this. Jesus is teaching us how difficult it is to judge properly. Comparing judging with eye surgery is very apropos. The only people that should judge is someone who can see clearly. The problem is that most of us lack in this ability. If I wanted a surgeon to do eye surgery on me, I would definitely look for one that is trained, and specializes in this field. Who is really qualified to judge? We often times judge on the limited knowledge we possess, and sometimes our own prejudice gets in our way of seeing clearly.

Finally apostle Paul reminded us in his discourse on love “For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known” (I Cor. 13:12).

I am aware that the scriptures teach that “the spiritual judges all things” but the key word here is things, not people.

τί δὲ βλέπεις τὸ κάρφος τὸ ἐν τῷ ὀφθαλμῷ τοῦ ἀδελφοῦ σου, τὴν δὲ ἐν τῷ σῷ ὀφθαλμῷ δοκὸν οὐ κατανοεῖς

Why moreover do you look the speck that in the eye the brother of you the but in to the your eye beam not consider up and down to a conclusion

This passage is specifically talking about the "brother" of us. Yeshua is asking why we clearly and quickly see a small little speck of unrighteousness in our brother, but we don't even consider the massive unrighteousness that we commit. What this means is, if you desire to judge your brother, consider your own sin very intently. By the time you're done, you should realize that you never have a right to judge anyone for anything.

However, this verse has nothing to do with those that do not consider themselves your brother or a son of God. Yeshua addresses non-believers in the last verse:

"Give not that which is holy unto the dogs, neither cast ye your pearls before swine, lest they trample them under their feet, and turn again and rend you." Matthew 7:6

In this verse, Yeshua is saying not to throw the precious word of God toward unbelievers. They could care less, you'll make them angry, and they may trample on the word of God and then beat the crap out of you.

Context is important.

Matthew 7:1-6 ESV Judge not, that you be not judged. For with the judgment you pronounce you will be judged, and with the measure you use it will be measured to you. Why do you see the speck that is in your brother's eye, but do not notice the log that is in your own eye? Or how can you say to your brother, ‘Let me take the speck out of your eye,’ when there is the log in your own eye? You hypocrite, first take the log out of your own eye, and then you will see clearly to take the speck out of your brother's eye. Do not give dogs what is holy, and do not throw your pearls before pigs, lest they trample them underfoot and turn to attack you.

Jesus is talking about judging others

Specifically when and when not to do it. The important line is "For with the judgment you pronounce you will be judged, and with the measure you use it will be measured to you" Examine your own life before pointing out sins of others. If you condemn others because of their sins when you're just as guilty it'll be like tossing pearls at pigs. The pigs see the pearls as worthless rocks and they think you're attacking them, so they turn and attack you back. This will happen even if your condemnation of their sin is correct.

The mote represents your brothers sins and the beam represents your own, which, when looked at very closely, are just as big or bigger than your brothers. The parable is saying to take care of your own sins and bad habits before trying to address your brother's.

Jesus' message is a pretty simple one, although we can glean a deeper lesson from it by paying attention to every word in his message. In modern parlance, Jesus' message could be paraphrased, loosely, as follows:

Get your own act together before criticizing someone else.

A mote is a speck of dust. A beam is a log or a piece of lumber used in constructing ceilings and roofs. The former is almost infinitesimally small, whereas the latter is gargantuan. The humor in Jesus' comparison becomes readily apparent in the contrast between the two.

Put differently, Jesus is saying,

"After addressing and dealing with your own hang-ups and besetting sins you'll be in a better, humbler position to confront a brother about his hang-ups and sins."

One of the often overlooked terms in Jesus' teaching in this passage is the word hypocrite (see v.5). A hypocrite is a person who pretends to be something he isn't. Put differently, a hypocrite is someone who does not practice what he preaches. For such a person to confront and criticize someone else before taking stock of his own failures and sins is analogous to "casting the first stone" (see John 8:7).

After making a "searching and fearless moral inventory of ourselves," we are less likely to be so judgmental of others. Why? Because after conducting that inventory of ourselves we can see more clearly that

1. We are just as guilty and condemned before God as they are, whether they or we have committed one sin or a million sins. "All have sinned and fallen short . . ." (Romans 3:23). Sin entangles all of us, without exception (see Hebrews 12:1), and James reminds us that "we all stumble in many ways" (3:1, my emphasis). To forget this is the height of hypocrisy.

2. Our quickness to point the finger of blame at someone else can belie the presence of the very same (or similar) sin we are accusing our brother of. Again, this is hypocrisy.

3. Our tendency to detect a fault in a brother violates an important principle in Scripture; namely, "love covers a multitude of sins" (1 Peter 4:8; Proverbs 10:12 and 17:9). To cover a sin is not the same as to cover up a sin and pretend it never happened. No, God is our example in this. He does not cover up our sins, but he does cover our sins through the blood of the sacrifice of his Son, our Lord Jesus, who died for our sins (1 Peter 3:18).

4. There is a better way to confront someone caught in a sin. Paul summed it up in Galatians chapter 6:

   "Brethren, even if anyone is caught in any trespass, you who are spiritual, restore such a one in a spirit of gentleness, each one looking to yourself, so that you too will not be tempted. Bear one another's burdens, and thereby fulfill the law of Christ" (vv.1,2).

In conclusion, there is a need within every local assembly of Christians to confront known sin. As Paul points out, however, only one who is spiritual is to do the confronting. Even then, spiritual people need to proceed with caution, because they are just as susceptible to stumble as the person they are confronting.

Rather than leading with a holier-than-thou attitude, they are to lead with caution and an I'm-just-a-sinner-too attitude. Sin breaks the heart of God, and sin should break our hearts, too, whether it is in us or in someone else.