The Uterus is called '子宫‘ (Zi Gong) when we speak about the Western medicine organ or '胞宫’（Bao Gong) when referring to the Traditional Chinese Medicine concept. It literally translates to the 'Palace of child'.
The TCM concept of Uterus (Bao Gong) doesn't see it as an isolated Organ but also includes other reproductive organs such as fallopian tubes, ovaries and the cervix. Its function is to store and discharge the menstrual Blood, thicken the endometrium, start ovulation, store the Kidney Essence (Jing 精), receive the sperm and house and nourish the foetus until birth.
The Uterus is one of the six 'extra Yang organs' (Uterus, Brain, Marrow, Bones, Blood Vessels and Gallbladder).
The Directing Vessel (Ren Mai 任脉) and Penetrating Vessel (Chong Mai 冲脉), which are among the 8 extraordinary Vessels, run through the Uterus and help it perform menstruation, conception and pregnancy. The Directing Vessel brings Qi, Yin and Essence and the Penetrating vessel brings Blood to the Uterus.
The Uterus connects to the Kidneys via the Uterus Channel. The Kidney Essence is the origin of menstrual Blood (also called Tian Gui in Chinese Medicine). It also plays a critical role for sex and conception of a child.
In terms of other Organ relationships Liver Qi stimulates menstrual Blood movement during the premenstrual phase. The Spleen, because it creates Blood from Food and drinks, also has an influence on menstrual Blood. The Stomach connects to the Uterus via the Penetrating Vessel and can therefore also impact it or be impacted by it.
The Uterus regulates menstrual Blood. The two most important Organs in assisting in this function are the Kidneys and the Liver. The Kidneys' Essence is the origin of menstrual Blood (also called Tian Gui). Liver Qi stimulates menstrual Blood movement during the premenstrual phase.
As such any Deficiency in the Kidneys or the Liver will inevitably lead to issues with menstruation.
During pregnancy, the Uterus carry and supply the foetus. To achieve this, it is essential to have Kidney Essence (through the Penetrating and Directing Vessels) and Blood (through the Penetrating Vessel) to nourish the Uterus.