“Don’t judge”. It’s something you hear all the time. At best, it’s misleadingly un-specific.
We must judge – we’re called to judge (John 7:24, Alma 41:14, Moroni 7:15-16). One of the most important things we must learn to do in this life is to judge correctly between good and evil.
The scriptures which say not to judge are also sometimes un-specific. But when they’re talking about judging people, the meaning is final judgement – deciding in your mind where a person is going eternally, and condemning them to that here and now in your actions towards them. This is unjust, because we’re not called to this type of judgement – even the Saviour said, during His mortal ministry, that He wasn’t called to give such judgement at that time (John 12:47). As I understand it, this is reserved for the moment when we each stand before God to receive our final judgement, based on our mortal experience.
So we need to be careful about how this injunction is expressed: rather than ‘don’t judge’, judge justly is more correct. At the right time, with the right motivation (love and clarity), with the right authority (it depends on the situation – some are ‘judges in Israel’, like bishops, and must judge those in their care on behalf of the Saviour, and in that way – so they can repent and come to Christ), and in the right way. This sort of judgement takes much effort, care, and thought. It requires a lot of us. Not taking this care is unloving and lazy – in both directions (judging and refraining).
Holding up my hands and saying, “I won’t judge you”, in this way, is refusing to care about a person enough to clarify something they might need, and neglecting their spiritual well-being. If you love someone as Christ does, then you will, like Him, help them see things they need to see in order to evolve. And you’ll do it in His way – with compassion, insight, and directness. In this area, the counsel in Doctrine and Covenants 121 is invaluable.
No power or influence can or ought to be maintained by the priesthood [and by any other means, such as friendship], only by persuasion, by long-suffering [(forbearance, patience)], by gentleness and meekness, and by love unfeigned;
By kindness and by pure knowledge, which shall greatly enlarge the soul without hypocrisy, and without guile –
Resproving betimes [(before it is too late/in good time)] with sharpness [(clarity)], when moved upon by the Holy Ghost; and then showing forth an increase of love toward him whom thou hast reproved, lest he esteem thee to be his enemy;
That he may know that thy faithfulness is stronger than the cords of death.
Doctrine and Covenants 121:41-44
Imagine if all interactions and judgement were done like this!
These are perfect instructions for speaking to someone about sin or weakness that endangers them or others. As I said, it’s a lot of work. Not doing it is letting a person go on in sin that might destroy their soul and those of others. Loving them – and anyone who will be influenced or affected by what they do – means doing something about it. To judge in this way is to love. Jesus pointed out to the ‘woman at the well’ the things she was doing that were wrong. She knew what they were. The gentle, direct way He did it helped her see herself and her life with clarity, and to desire cleanliness. He didn’t sit with her and say, “Oh, you’re a lovely person; I know you only have good desires; I’m not going to say anything about what you’re doing; let’s talk about the weather.” He spoke about exactly the thing she most needed to hear. He offered her the water of life: repentance, forgiveness, and holiness.
We, as Christ’s followers – and His messengers to the world – are to do the same. As the Saviour said to Moroni, when people come to Him, He shows them their weakness. If He didn’t, nothing would happen. We wouldn’t change, we wouldn’t evolve, and our lives here would be wasted. Therefore, if we love others, we’ll lead them to Him and not shy away from clarity and truth, wherever they’re needed. And we’ll do it with the spirit of Christ – of love, deep compassion (the kind that cannot hold back), clear knowledge, and patience to help them along the way – and meekness to receive the same when we need it.
So the injunction to “judge not”, in the sense of judging another person’s actions, etc., should be taken to mean, “take care that you judge only in this way – and if you won’t, don’t judge a person’s situation at all.”
Of course, this isn’t the only type of judgement we need to do here: there is also judging for ourselves, and discerning another person’s intentions, as well as judging their actions in order to determine whether to believe or follow them, whether their influence is helpful, and other such things. These are important judgements that we must make.
Let us love more by judging more clearly and carefully. Let us rock the boat, whenever and wherever it’s needed – in the most careful and loving way possible.
Peaceable Treasures 
In other words, we can and should judge actions, rebuking others if necessary. But judging someone’s place before God is another matter. One should also not judge rashly.