What is the meaning of the term "Ma Malakat Aymaanikom"? And what kind of relationship is permitted with them?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

As for the meaning of the term "Ma Malakat Aymaanukom" (i.e. the slaves that your right hands possess), it is mentioned in the Qur'an in many places (verses).
Allah Says (interpretation of meaning): {And those who guard their chastity (i.e. private parts, from illegal sexual acts) Except from their wives or (the captives and slaves) that their right hands possess, for then, they are free from blame;} [23: 5-6].
This term includes the slave girls and slaves in general those who are under control of a free Muslim. As a rule, the only channel of producing this segment of society is Jihad in the cause of Allah. Slavery was common without control before Islam in all nations. So, Islam retains this system as a rule, but sets regulations that control, govern and try to root out such a system. Among these regulations are:
1) Islam urges that slaves be freed as an expiation for unintentional killing, a broken oath or, stating al-Zihaar to his wife (refusing sex with his wife, claiming she is like his mother).
2) Islam urges that Muslims treat their slaves politely, to educate them, to not burden them with tasks, and to help them if needed.
3) Islam prohibits selling a free person and taking the price, and confines slavery to Jihad, i.e. captives become slaves only by Jihad.
Thus, we state that in Islam slaves are treated with leniency and kindness, since the slave comes to know Islam, its good morals and tolerance. This leads him to embrace Islam, the only path of guidance.
At last, a Muslim has the right to have sex with a slave girl since she is "in the possession of his right hand". Then, if she has a child, it becomes Haram to sell her, and when her master dies, she becomes free.
Allah knows best.