Most people who are interested in mindfulness know that it comes from Buddhism. Fewer know, however, how far the modern, popular conception of mindfulness diverges from what the Buddha probably meant by it, 2,500 years ago. And fewer still know how useful the Buddha's mindfulness might be to us today.
The modern conception of mindfulness is pretty well summed up by Jon Kabat-Zinn, its most successful popularizer. He said that mindfulness is "paying attention in a particular way: on purpose, in the present moment, and nonjudgmentally".
And it's true to say that this vision of mindfulness is derived from Buddhism. But Buddhism is a large and varied thing, with many schools and traditions that have evolved over millenia, many of which diverge substantially from the original Buddhism of the Buddha (which wasn't called "Buddhism", by the way, until Western colonists came along). So, Kabat-Zinn was drawing on the traditions that he had been trained in: Korean Zen (called Son), and modern South East Asian Vipassana. There's nothing at all wrong with that, and his presentation of mindfulness has been of great benefit to many people, me included. But there's also benefit in exploring the earlier meaning of "mindfulness".
In fact, we don't know for sure what the Buddha himself meant by "mindfulness", because the earliest Buddhist texts date from a few hundred years after his death, and aren't even written in the language that he would have spoken. But those texts are the closest we are going to get to the thought of the Buddha himself, and they make reference to something called sati, which is what gets translated as "mindfulness".
Sati didn't mean a non-judgemental awareness of the present moment. Instead, it was something closer to "remembering", or "heedfulness" - an awareness that held in mind all of the Buddha's teaching, and continually applied it to whatever was happening in the present moment. So, it was very much about judgement: you would notice what you were thinking and doing, and evaluate it according to the many principles, rules, and recommendations that the Buddha offered. Sati was a way to stay close to what was wholesome, and to avoid what was unwholesome. It was certainly not a way to put aside all judgements and simply rest in the here-and-now.
This might not sound terribly appealing to us moderns. Who wants to remember a load of rules and recommendations about what you should and shouldn't do, and still less be bound by them? But in fact this view of mindfulness addresses two major reservations that we might have about the modern version of mindfulness. The first is: But don't I need to think? How will I get anything done if I let go of all thoughts and judgements and just rest in the present moment? And the second is: What about morality? Don't I need to judge, to know what's right and wrong?
One sensible answer to these is: Don't worry, there is really no danger that you are going to lose your ability to engage with and act on your thoughts. Rather, learn to connect more fully and more often with the present moment and you'll gain a little more control over when you engage with and act on your judgemental thoughts. Worrying that you'll lose the ability to do so altogether is like avoiding the gym because you don't want to look like a world-champion bodybuilder - it's just not going to happen. Your training in mindfulness or in the gym isn't going to be that effective.
But, nonetheless, these are not completely implausible pitfalls. Some people really do use mindfulness to avoid shame and regret about unwise or immoral actions, reminding themselves that regrets are "just" thoughts and feelings. One wonders if some spiritual teachers have done this, while abusing their flocks in various ways. And the more you focus on the present moment and resist entaglement with thought, the harder it is to e.g. remember your keys when you leave the house. (During my initial honeymoon period with mindfulness, I almost locked myself out in just this way.)
Most striking, someone once published a study of people who claimed (and were considered by others) to be enlightened: https://nonsymbolic.org/PNSE-Article.pdf. One of the themes that emerged was that these people a) reported experiencing far fewer thoughts than they had before (supposedly) becoming enlightened, and b) had absolutely terrible prospective memory - the kind that reminds you to pick up your keys and do the laundry. Their brains had stopped producing thoughts like, "Remember your keys", and, "Time to put on the laundry", and so their houses were covered in post-it notes reminding them to do those things.
So, maybe the Buddha's view of mindfulness continues to have value. We need to think and to act on our thoughts in order to function in the world, and we need to discriminate between what is a good idea and what is not. And we also need to remain tethered to the present moment: over-involvement with our thoughts is generally painful and stops us responding appropriately to what is actually happening, here and now. What is optimal, most of the time, is something very like what the Buddha prescribed: a keen awareness of your own mind and body and your surroundings, and also a recollection of your intentions or values - the principles that you would like to guide your behaviour.
To cultivate this kind of mindfulness, try this. Think of one of one or more things you would like to do, or refrain from doing. Perhaps you'd like to be more conscientious at work. Or perhaps you'd like to stop snapping at your partner. These are your intentions. At the beginning of your morning meditation, or just at the beginning of your day, take a few moments to reflect on those intentions, and commit yourself to them. Commit fully to holding them in mind and acting on them, as best you can. Try and really mean it. And then do it. As you go about your business during the day, maintain your awareness of what is happening, right here and now, but also of your intentions; of how you want to act. Remain mindful of what you're doing, and also of what you want to be doing, and mindfulness can start to change your life.