Broadly, I like to categorize breathing meditation into two types.1 The first involves active breathing. You breathe in, you breathe out. Sometimes you count while you breathe, sometimes you hold your breath, sometimes you picture emptiness while you breathe out. Whether you are monitoring the sensations that accompany the breath or other mental visualizations, the key is you are actively controlling and modulating your breath during these meditations, willing it as you go.
There is a second kind of that involves passively observing your breath.2 In our daily lives we mostly don't think about having to breathe, and we breathe anyway. That is what you monitor — when did your body and brain decide to inhale? To exhale? Did monitoring your breath perturb your natural rhythm just now? You realize that, just like you don't control when sounds or sights enter your mind, there are times when you also don't control when that impulse to breathe comes and goes, and those impulses are things you can monitor and "receive" as they come and go.
An interesting thing about breathing is how you can fairly seamlessly transition between controlling the breath and letting it do its own thing.
I was trying to snorkel with a shoddy set of snorkels at Makena beach in Maui. Shortly after plunging my face under water, in a few moments I could taste the spray and hear the gurgle of ocean water in my snorkel mouthpiece. It was annoying but I didn’t think it was necessarily very much, so I tried to breathe in anyway so I could stay under water to look at the reefs and continue swimming.
To my surprise, as much as I willed I couldn't. Something was holding me back from breathing. So I just held my breath for a bit longer, regrouping before trying another time. And again, as much as I willed to inhale I couldn’t. So I got up, spit out the snorkel, took a breath above water, before plunging down, and trying again.
My breathing reflexes did not overwhelm me in a panic to force me above water — I could still stay underwater for longer if desired. But neither could I force myself to breathe. I'm not sure the last time I was confronted with such a striking limbo state between the conscious and the unconscious, what can and can’t be willed. I suppose it’s not unlike sleep paralysis and I bet it happens in many other contexts.
For some more specific introductory breathing meditation, see this article. There are actually suttas on breathing as part of mindfulness, though they need helpful commentaries to elucidate.
Although someone might notice that this can be an active decision to “step back” and observe. In fact, it might even take a lot of training and effort to be able to simply step back and observe. My understanding is that with enough repetition some people actually default to an open, observational state. This is a long way of saying that “passively observing breath” doesn’t necessarily mean a complete absence of intention, effort, or activity.