You observe, without passing judgment. If a thought comes, just let it come, and don't 'reply' to it. When you notice you drift off, just acknowledge that, without judgment, and continue the process of observation.

For the sake of clarity, let's look at the mind as if it's a separate entity (like a child) that is constantly asking our attention. When we give in, when we reply to our thoughts, its behavior is rewarded and it will keep asking for attention. If you just acknowledge that it's there, but don't act on every thought, the stream will become less and less. The main goal being that it eventually shuts up unless you actually need (ask) your mind to help you with a task.

Other comments have already talked about what mindfulness meditation is but that's only a part answer.

You could say it's not different, since anything done in that physical pose is something we're going to call meditation. There are a lot of different practices from different cultures, and really what they have in common is that you quietly sit and focus on being in a certain state of mind, continuously trying to bring yourself back to that state of mind if/when you get distracted.

For mindfulness it's non-judgemental awareness of the present moment and your own mind's activity

For zen it's controlled focused attention on the object of meditation (nothing at all in the cliche)

For compassion meditation it's compassion/kindness/love towards yourself/others/everything

For the various types of body awareness meditation it's your own feeling of your physical body

There are mantras and guided meditation and many others, with a lot of historical cross pollination and tons of wacky woo-filled stuff. Many types that are likely complete junk that's useless or harmful because the whole edifice is built on magical thinking.

I think the feasible technical explanation is pretty simple, "neurons that fire together, wire together". It seems very reasonable to me that practising attaining a certain state of mind will make it easier to do at will and/or make shift your normal state in that direction. A lot of it is literally as simple as learning "If I focus on X for a short time I feel like Y, so now I know how to feel like Y when I want"

There are many observable phenomena for which we lack a complete technical explanation.

You are actively trying to change how you think. It's not necessarily thinking about nothing. In many cases it's about coming at the same thoughts from a different perspective, often an external and non-judgemental one.

It really depends on which tradition or practice of meditation you're talking about. In the exercise I go through, it's different because it's specifically focused on acknowledging whatever thoughts come through without judging or reacting to them like I would impulsively. Being able to accept especially the negative thoughts at face value helps me a lot with managing the coping process.

My understanding is that you're supposed to let things come, but to note their happening in your mind's voice. I start by noting each breath, then as my awareness of it fades in and out I note people, noises, feelings, and thoughts.

In my humble opinion / understanding, mindfulness is about reaching a state where you observe what's happening around you without judgment. When you begin actively pushing thoughts away, you're once again engaging the ego.