What is this called?
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“抽屉”：可能是橱或柜的一部分，可以抽放（Maybe a part of cupboard or cabinet we can pull it out and put things there and push it in）.
If we see a thing full of “抽屉”, a collection of them one beyond one together (like the image you showed us). This is called “柜”. This may be high or low, but most cases this is low or mediam of height:
And for most cases, if a high thing full of small doors together for us put bags, books……ect. This is also a “柜”：
In the acient China, “柜” was something like a box (can open or close, horizonal and movable).
And if you see a high thing with a door open, where there's a large space seperated by thin boards to put cups, glasses, cropsticks or in a room for clothes…… with no 抽屉, this should be “橱”.
HOWEVER, with the rapid development of our cupboard, we make "橱" and "柜" together called "橱柜": This is a thing (most case this takes after "橱", with a few 抽屉), we usually use "橱" for glasses/cups……, “柜” for kitchen small things like scissors，rag……).
All in all, for "clothes" or "cups", we usually put them into the “橱”, the others are usually put in the 柜.