A happy or warm feeling in the chest, usually in response to
prayer, that
Mormons consider a witness from
God and take to be
evidence of
truth.
A fundamental concept in the gospel of Mormonism is that God reveals truth to those of his followers who ask for it. Mormons are taught that the most common way he does this is through feelings when asked in prayer if something is true. Truth manifested this way is taken to be revelation directly from God and is therefore considered superior to truths acquired through secular means.
An example: Mormon missionaries commonly invite a person to read the Book of Mormon and then ask God whether it is true. If an investigator experiences feelings of peace, happiness or warmth after praying he is told this is God's message to him confirming the truth of the book.
In Mormon scripture written in the 1830s these feelings of peace, especially if physically manifested, were referred to as "burning in the bosom", and this term has crept into colloquial usage. It sometimes provides for hilarity given the double entendre possible with the term "bosom" in modern times.
The opposite of a burning in the bosom is a "stupor of thought", whereby you can know that the subject of your inquiry is false or not of God. Mormons are advised to use this method (prayer followed by the close examination of one's feelings) as a guide for assessing religious truth, but also even for help with making some of life's most mundane decisions, such as which school to attend or whether to take a new job.