Vase breathing (Tib. རླུང་བུམ་པ་ཅན།, Wyl. rlung bum pa can) is so called because it involves holding or 'containing' the breath in a way that is analogous to a vase or container.
There are two quite different practices called vase breathing, and the differences between them are significant. There is the vase breathing practice associated with chandali or tummo (inner heat) meditation, and there is the vase breathing practice associated with tranquillity meditation or mantra recitation.
When you practise vase breathing as part of chandali or tummo practice, because your intention in that practice is to generate physical heat or bliss, the vase breathing needs to be quite intense and energetic, whereas when used during meditation or mantra recitation it is more relaxed.
- *Thrangu Rinpoche, The Ninth Karmapa's Ocean of Definitive Meaning