Breathing is quite important and since it is something that we do from the moment of birth to the moment of death, you might think it's a pretty straightforward thing. Breathe in, breathe out, repeat... But the more you investigate it, the more complex it becomes. There are LOTS of different ways to breathe and they have a wide variety of different effects and purposes.

You can breathe in and out through the nose or through the mouth or in through one and out through the other. There are also Yoga exercises where you breathe in through one nostril and then out through the other one.

I generally breathe in and out through the nose when doing Tai Chi and the same for most of the Qigong exercises that I practice, there are however some Qigong exercises where 'in through the nose and out through the mouth' is the way.

For health and wellness practice, I generally breathe in when I am doing any movement that naturally expands the body and out when the movement pulls the body inwards. However that is not the case for martial Tai Chi where you generally breathe out during any movement that represents you hitting or pulling or pushing another person.

As to the benefits of the different breathing methods, that is possibly up for debate, but there is an article on 'nose breathing' at: https://www.healthline.com/health/nose-breathing
I find it very interesting where they say: "During nasal breathing, your nose releases nitric oxide (NO). NO is a vasodilator, which means it helps to widen blood vessels. This can help improve oxygen circulation in your body."

So, something that we generally give little thought to, is in fact, worthy of some study and practice.