Its the fighting styles from the series.

[quote]Breath Styles make use of specific and concentrated breath patterns, which increase the users lung capacity and amount of oxygen in the blood.

The increased amount of oxygen allows the user to increase their physical abilities and mental concentration to the point where they can match a Demon's strength. Focused breaths are also capable of clotting wounds from severe injuries and slowing down the circulation of poison in the bloodstream.[/quote]

From the wiki.

Essentially they are your characters fighting style but do have a bit of lore to them.