In Islam a Muslim is forbidden to marry a disbeliever:
And do not marry polytheistic women until they believe. And a believing slave woman is better than a polytheist, even though she might please you. And do not marry polytheistic men [to your women] until they believe. And a believing slave is better than a polytheist, even though he might please you. Those invite [you] to the Fire, but Allah invites to Paradise and to forgiveness, by His permission. And He makes clear His verses to the people that perhaps they may remember.
However, Christians and Jews have different rules. The majority of scholars say that Muslim men are allowed to marry Christian and Jews in accordance with the actions of the Prophet, his companions and followers and the following Ayah:
This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them. And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers.
The final point which many will argue against is a Muslim woman marrying Christian/Jew. Sadly there is no proof from Quran and Sunnah to back that up, but still many scholars would say it is not allowed. Their main proof is the first Ayah I posted, which any smart person can clearly see that it doesn't apply since Christian and Jews are not considered disbelievers.
All hadith's and Quran Ayah's that have been referenced to such prohibition are from Muslim woman who enter Islam and the Prophet prohibits them from returning to their hubbies when they are disbelievers (mushkreen) it was NEVER reported that he did the same for those whose hubbies were Christian or Jew.
It was reported by Jaber, that the prophet said: We marry the woman of the book but they don't marry our woman.
عن جابر – رضي الله عنهما – قال : قال رسول الله صلى الله عليه وسلم : نتزوج نساء أهل الكتاب ولا يتزوجون نساءنا
However, this hadith was classified as VERY weak by most scholars because:
Another invalid proof was the saying of Omar ben Alkhatab when he said: The Muslim man marries a Christian but the Christian can't marry a Muslim woman.
وى ابن جرير الطبري في تفسيره (4/366) بإسناده عن عمر بن الخطاب – رضي الله عنه – أنه قال : المسلم يتزوج النصرانية ، ولا يتزوج النصراني المسلمة .
Sadly this is another MAJOR weak hadith in accordance with Alhafid:
والأثر في إسناده يزيد بن أبي زياد الهاشمي ، قال الحافظ ابن حجر في التقريب : ضعيف كبر فتغير وصار يتلقن وكان شيعيا .
As you can see, my opinion is that Muslims men alike are allowed to marry the people of the book (Christians and Jews) regardless of believing in the trinity, son of God, etc.. as most of them believed during the time of the prophet and his companions and that didn't stop them from marrying them or eating their food.
Also as you can see, part of my opinion is that Muslim woman can also marry Christian/Jewish men as proven by my argument above and NO other proof contradictory to that opinion from the Quran or Sunnah.