There are differences in opinion as to what is regarded as halal. This depends on which madhab(school of law) or scholarly opinions you follow.
It is clear that no meat can be consumed without saying Bismilah. According to Sh. Yusuf Qaradawi, he said the following.
"It is not required of the Muslim to inquire about what he has not witnessed, i.e., How was the animal killed? Did the manner of slaughter meet the Islamic conditions? Was the name of Allah mentioned while slaughtering or not? If the animal was slaughtered by a Muslim, even if he is ignorant or sinful, or by someone from among the People of the Book, eating it is halal for us."
The Prophet was asked, “People bring us meat and we do not know whether they have mentioned the name of Allah over it or not. Shall we eat it or not?” and the Prophet (peace be on him) replied, “Mention the name of Allah (over it) and eat.” (Bukhari)
Concerning the application of this hadith, scholars say: This is proof that the actions and practices of people are ordinarily considered to be correct and appropriate, while deviation or error must be proved.
Concerning meat from the Christians and the Jews.
According to Sh. Yusuf Qaradawi, meat from Christians and Jews are halal for Muslims. He quotes the verse Quran 5:5. He states that while Islam takes an uncompromising attitude towards polytheists, it is lenient towards the People of the Book for they are closer in their belief in divine revelation.
Sh. Yusuf Qaradawi also states that even animals slaughtered for Christian festivals is halal for us to eat if we don't hear the name of other than God mentioned. He also mentions that some jurist are of the opinion that because food from the People of the Book has been permitted to us by God, then so is anything that is slaughtered by Christians and Jews regardless of whose name is mentioned.
Narrated by At-Tabari, someone asked Abu ad-Darda' whether he could eat the flesh of lamb slaughtered for the Church of St.George, which has been given to him. Abu ad-Darda' answered, "Are they not the People of the Book, whose food is halal for us and ours for them?" He then told the person to eat it.
Imam Malik was once asked about eating the flesh of animals slaughtered for Christian festivals and churches. He replied, "I classify it as mukruh but not haram."
He classified it as makruh because he was afraid that it might have been dedicated to someone other than Allah but not as haram because perhaps, with respect to the People of the Book, the meaning of the phrase, 'that which has been dedicated to any other than Allah,' applies only to those animals which they slaughter for the purpose of seeking the pleasure of their deities and not to eat. As for what they slaughter to eat, it is their food, and God says, "The food of those who were given the Scripture is permitted to you...." (Quran 5:5).
This has been taken from 'The Lawful and The Prohibited in Islam' by Sh. Yusuf al-Qaradawi. Pg. 56 - 58
An important point to take note
The first answer provided by islam101 claiming that if one does not eat halal, our duas will not be answered. This is incorrect. The hadith islam101 used to justify this is from the book of zakat. This hadith refers to having a legitimate income and not about the consumption of halal food. If one's income is unlawful such as gaining financially by forcing someone into prostitution, then his food is unlawful, his cothes are unlawful and his nourishment becomes unlawful. How can Allah accept his supplications?
If this hadith was about the consumption of halal food and the effects of supplications, then this hadith would be in the book of supplications or the book of dietary requirements.
And Allah knows best.