For Muslim Man
A Muslim man is allowed to marry a non-Muslim women. If she is Christian or Jewish, other than these two religion marriage is not allowed. In Quran Allah says,
“Made lawful to you this day are At‑Tayyibaat [all kinds of Halaal (lawful) foods, which Allah has made lawful (meat of slaughtered eatable animals, milk products, fats, vegetables and fruits)]. The food (slaughtered cattle, eatable animals) of the people of the Scripture (Jews and Christians) is lawful to you and yours is lawful to them. (Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture. (Jews and Christians) before your time when you have given their due Mahr (bridal-money given by the husband to his wife at the time of marriage), desiring chastity (i.e. taking them in legal wedlock) not committing illegal sexual intercourse, nor taking them as girlfriends.” [al-Maa’idah 5:4]
But a Muslim man cannot marry a idol-worshiping women or a communist women. In Quran,
“And do not marry Al-Mushrikaat (idolatresses) till they believe (worship Allah Alone). And indeed a slave woman who believes is better than a (free) Mushrikah (idolatress), even though she pleases you.” [al-Baqarah 2:221]
A mushrikah speaks of those who commit ‘Shirk’. Mushrikah is one who worship idols and stones which is Shirk. In Quran the word ‘Shirk’ means to associate partners with Allah and whosoever do this is not a Muslim and thus marriage is not allowed with them.
For Muslim Women
A Muslim women is not allowed to marry a non-Muslim from any other religion including Christians and Jews. Quran says,
“And give not (your daughters) in marriage to Al‑Mushrikoon till they believe (in Allaah Alone) and verily, a believing slave is better than a (free) Mushrik (idolater), even though he pleases you. Those (Al-Mushrikoon) invite you to the Fire, but Allaah invites (you) to Paradise and forgiveness by His Leave, and makes His Ayaat (proofs, evidences, verses, lessons, signs, revelations, etc.) clear to mankind that they may remember.” [al-Baqarah 2:221]
Allah didn’t allow Muslim women to marry non-Muslim men including the People of Book, because women are usually follow the culture and tradition of their husbands. When a women marries any man then she is expected to worship the way her husband and his family does. Thus, she automatically become part of their religion and in the same way the children that she bring to life will also follow the same religion and will also be non-Muslim. Marrying a non-Muslim man would effect the Muslim status of women. Thus she is not allowed to do so.