Can our minds pay attention without judging?

With practice, yes. Mindfulness is about focusing on the present moment, and accepting whatever you happen to find in the present. Acceptance - or being nonjudgmental - is a challenge; think of how most of us grow up, constantly hearing judgments: "that's good" or "that's bad"; you're "right" or "wrong", "what were you thinking??" and so on. Our society is a judgmental one, and growing up in a judgmental society means that we usually are quite judgmental ourselves.

While it is a challenge for our mind to pay attention to something without judging it, it certainly is possible with a lot of practice. Start by just noticing your judgments; often we can actually reduce an unwanted behavior just by paying more attention to it. Once you start to notice your judgments, you can work on changing them into nonjudgmental or neutral statements, by sticking to the facts of the situation, and expressing your emotions about the situation (it's important to remember that most of the time we judge because we're feeling some kind of painful emotion, such as anger, hurt, fear, etc.). Think of being nonjudgmental as being assertive: you're still expressing your thoughts and feelings, but you're doing it in a nonjudgmental, non-blaming way.

So remember: the first step is increasing your awareness. Just make sure that, once you become more aware of your judgments, you don't judge yourself for judging - remind yourself that it's normal, your human...AND that this is a behavior you want to reduce. Reducing your judgments, and bringing more acceptance into your life, is a healthier, happier way of living; so it's worth the hard work!