What is the name of part of onion we peel?

We call the state you can’t stop tearing “玉ねぎの皮をむくように―tamanegino kawa wo muku youni – like peeling the ‘skins’ of onion" in Japanese. In actuality, we don’t shed (or drop) tears when we peel off the outer skin of onion. We shed (or drop) tears when we peel the inner layers of onion.

This might look a very primitive question to Anglophones. But none of Japanese English dictionary at hand carries English counterpart to ‘the inner layer’ of onion. It can't be flesh, pulp, capsule, or leaf. No English text books available in this country makes a specific mention to the inner part (layers) of onion.

What is the exact word for the thing (part) of onion we peel off in colloquial English, and if possible with botanical nomencclature?

They are called scales - outer ones are membranous and inner ones fleshy. The protective thin outer covering is called tunic.

http://www.taylor.k12.ga.us/~tcms/onion/images/onionparts

We peel the skin off the onion, then after that, we peel layers of the onion.

The onion is a bulb.

The skin is the membrane on a layer.

Each layer has a skin, a membrane that wraps around the actual layer.

The layer itself is actually a leaf.

The very center does not have a name since it is just younger leaves growing out of the basal disc.

Once you get past the thin, dry, brown skin, the onion is composed of many layers.

We often use onion or peeling an onion as a metaphor for something that has many layers. For example, there is a system for browsing the web anonymously called Tor. Tor is an acronym for The Onion Router, which refers to the fact that all communications are wrapped in many layers of security.