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'I will breathe in tranquilizing the bodily formation'; he trains thus: 'I will breathe out tranquilizing the bodily formation.'" ---Satipatthana (Bhiku Bodhi's translation)
What is bodily formation? Does it mean tranquilizing the breath or the body?
OP: What is bodily formation?
Kaya Sankhara - this what sustains keeps the body together hence the breath. If breathing stops the body falls apart.
OP: Does it mean tranquilizing the breath or the body?
To do this, you should decide to make the breath tranquil, and go on being continuously aware of the breath from beginning to end. You should do nothing else, otherwise your concentration will break and fall away.
The Visuddhi·Magga gives four factors for making the breath tranquil:
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Concern (ābhoga): you pay initial attention to the breath, you apprehend the breath, you advert the mind towards the breath, to the effect: ‘I will try to make the breath tranquil.’
Reaction (samannāhāra): you continue to do so, i.e. you pay sustained attention to the breath that way, do it again and again, keep the breath in the mind, to the effect: ‘I will try to make the breath tranquil.’
Attention (manasikāra): literally ‘deciding to make the breath tranquil’. Attention is the mental factor that makes the mind advert towards the object. Attention makes the mind conscious of the breath and know the breath.
Reviewing (paccavekkhaBa): you review (vīmamsa) the breath, make it clear to the mind, to the effect: ‘I will try to make the breath tranquil.’
p35/36 Knowing and Seeing by Pa-Auk Tawya Sayadaw
Kayasankara is the breath (MN 44).
Kayasankhara is properly translated as ‘body conditioner’, similar to the meaning of ‘body sustainer’ or ‘body nourisher’.
The 3rd step of Anapanasati is properly translated as 'experiencing all bodies'. There are three 'bodies' or 'kaya', namely, breath body, physical body (rupa-kaya) and mental body (nama-kaya). In the Anapanasati Sutta, the Buddha says: "the breath is a body among other bodies". In brief, the 3rd step is to experienced how the three bodies interrelate with or inter-condition each other. For example, when the mind is calm, the breath will be calm, then the body will be calm; if the mind is agitated, the breath will be agitated; then the body will be agitated.
The 4th step of Anapanasati is training how to use the mental body (mind) to tranquilize the breathing body; which in turn tranquilizes the physical body; which in turn tranquilizes the mind and eventually makes the mind feel happy (step 5 & 6, rapture & happiness).
In summary, tranquilizing the kayasankhara means tranquilizing the breathing, which will simultaneously tranquilize the physical body.
Refer to Bhikkhu Buddhadasa's book: Mindfulness With Breathing - Unveiling the Secrets of Life: a Manual for Serious Beginners.
