Whatever it means to 'judge', this we are fully and absolutely forbidden from doing in Mat 7:1. The reason given is not there to qualify the judging but to support the prohibition.
A careful look at the entire sermon on the mount shows the meaning of the term and what is being prohibited: it is a) to sue in the state courts b) to judicially judge in such courts, and c) to enforce judgement debts by way of debtor's prison, debt-slavery, seizure of property, flogging, death penalty or the like. This can be seen from:
Mat 5:21-26 compares and contrasts two cases: the capital case of murder and the civil damages case of insult. Unlike the law of Moses, as the time Jesus taught capital punishment was de facto abolished: the judgement for murder was not carried out, by the Jewish authorities at least, the Romans had no objection and used it against trouble makers. Unlike the law of Moses, at the time Jesus taught, civil litigation for money damages compensating insult was recognised. Jesus is mocking the civil litigation system in this passage, portraying a man whose only injury is to his pride, filing suit, obtaining judgement, and enforcing the judgement debt to financially ruin the judgement debtor, putting him in debtor's prison until he has paid the last penny of the judgement debt. He also warns that if you insult the powerful people, you are in danger of capital punishment imposed by the Romans, putting your body in danger of dishonourable disposal in gehenna, the rubbish dump outside the city, where our Lord's body also nearly ended up, as a result of him insulting the powerful people.
Jesus also prohibited swearing oaths (Mat 5:33-37), thereby prohibiting both filing of suit and rendering of judgement, for no suit can be filed except supported by sworn testimony, and no one can be found liable or guilty except on the sworn testimony of two or three witnesses. Prohibiting the swearing of oaths also means we cannot do business or contract obligations with the implicit recourse to the coercion of the state courts.
The fundamental legal principle of civil litigation is seeking financial compensation for the damage caused by the fault or wrong of someone else. This principle Jesus repudiates by abolishing the 'eye for eye' administration of justice (Mat 5:38-42). In its place Jesus prohibits resistance to evil. Filing suit and awarding damages is a form of resistance to evil, but Jesus prohibits it.
The alternatives to filing suit are: a) rely on or appeal to honour and b) forgive the debt. Jesus counsels both in the sermon on the mount, indicating what he has prohibited.
In place of civil litigation in state courts Jesus provided an alternative procedure that amounts to appeal to honour in Mat 18:15-17. Note that the alternative procedure does not include sworn testimony, compulsory jurisdiction (dragging to court), debtor's prison, debt-slavery, seizure of property, capital punishment or flogging.
This alternative procedure and set of remedies can be called judging / suing and it is in this sense that we are not prohibited from doing it (1 Cor 5-6). The same word judge can also be used in the sense of discernment or discrimination, and this we are not prohibited from doing, either. Judgement of the person himself, from his own heart, is exactly what is called for in the absence of appeal to state court remedies. If we follow our conscience and seek honour before God rather than before men we shall not wrong our fellow man and shall willingly repent of our wrongs and pay our debts and compensation to those we have injured without even being asked. In fact we are called to use this discernment to judge the nature and effect of filing lawsuits and rendering judgement and enforcing judgements, as it is the sign of the evil times:
He said to the crowd: “When you see a cloud rising in the west, immediately you say, ‘It’s going to rain,’ and it does. And when the south wind blows, you say, ‘It’s going to be hot,’ and it is. Hypocrites! You know how to interpret the appearance of the earth and the sky. How is it that you don’t know how to interpret this present time?
“Why don’t you judge for yourselves what is right? As you are going with your adversary to the magistrate, try hard to be reconciled on the way, or your adversary may drag you off to the judge, and the judge turn you over to the officer, and the officer throw you into prison. I tell you, you will not get out until you have paid the last penny.”