Question: Can a Muslim woman marry a Non-Muslim man, or vice-versa?
Muslims are allowed, but not advised to marry with Non-Muslims. Allah the Almighty commands:
“Do not marry mushrik women (who interpose others between herselves and God) until they trust only in God. A captive woman who trusts only in God is better than a mushrik woman even if she (the mushrik) has impressed you much. Do not wed your women to mushrik men (who interpose others between himselves and God) until they trust only in God. A captive man who trusts only in God is better than a mushrik man even if he (the mushrik) has impressed you much. They invite you to the Fire but God invites you, by His leave, to the Gardens and to forgiveness. He manifests His verses to the people so that they may come to their senses.” (2:221)
As can be seen in the verse, Allah advises Muslims to marry Muslims, rather than mushrik people. This is not a prohibition. If something is defined to be “better than” the other, that doesn’t necessarily make the first one prohibited. There is only one prohibition for the Muslims about marriage, and that prohibition is “marrying with adulterous people”. Allah the Almighty commands:
“A man guilty of fornication shall not marry any but a woman guilty of the same or a mushrik woman, and none shall marry a woman guilty of fornication other than a man guilty of the same or a mushrik man. Such marriages are forbidden to the believers.” (24:3)
Please see the link below for further information:
 The word Mushrik is of the root shirk. Shirk, literally means “making partners” in Arabic. In terminology, it means “Neglecting Allah, or regarding other things rather than God in the first place while making a decision”. Anything put in the first place is considered by the person as an authority of the same level as Allah. Therefore, it is like a partner of Allah. Mushrik is the subject form of shirk.