Doctrine and Covenants (D&C) Section 9 contains one of the best known passages in Latter-Day Saint scripture.

7. Behold, you have not understood; you have supposed that I would give it unto you, when you took no thought save it was to ask me. 8. But, behold, I say unto you, that you must study it out in your mind; then you must ask me if it be right, and if it is right I will cause that your bosom shall burn within you; therefore, you shall feel that it is right. 9. But if it be not right you shall have no such feelings, but you shall have a stupor of thought that shall cause you to forget the thing which is wrong; therefore, you cannot write that which is sacred save it be given you from me.

This passage teaches the general principle that, if we want the Lord’s guidance in a choice, we should first use the intelligence and resources he has given us to make a thoughtful decision. Only then should we ask for divine confirmation of the correctness of our tentative decision. A correct decision is confirmed by a divinely induced burning in the bosom, while an incorrect one is signaled by a stupor of thought.

“What is a burning in the bosom? Is it a warm feeling in your chest? No.

The idea of one’s bosom burning had it’s peak in the late 1700s. Here are some uses of the phrase “bosom burn” from that period, from Google NGram Viewer:

“There shall my bosom burn with friendship’s flame” (1737)
“While truth and virtue in thy bosom burn” (1743)
“He thank’d her care; yet day by day His bosom burn’d to disobey” (1733)
“Honorius’ bosom burn’d with fierce desire” (1737)
“And love did with a chaster flame Within my bosom burn.” (1622)
“With sudden Grief her lab’ring bosom burn’d” (1745)
“By Tyrants first, then by a Brother spurns, Still, still, with Loyalty his Bosom burn’d” (1745)
“Whilst the bright Flames which in his Bosom burn” (1708)
“Genius of Britain ! bid those days return, For Thee, for Virtue, let each bosom burn.” (1780)
“If e’er thy bosom burn’d with lawless love, Art thou to pining avarice a slave?” (1790)
“The stage breath’d war— the soldiers bosom burn’d, And fiercer to the field each chief rerurn’d” (1764)
“With sacred Thirst my Bosom burn’d” (1788)
“With martial wrath his ardent bosom burn’d” (1795)
“Each bosom burn’d to meet their foes, and swore to save their native land.” (1798)

Your bosom is your innermost self, so a burning in your bosom is something you feel deep inside. Based on the examples above, that feeling can be friendship, truth, virtue, desire, love, grief, loyalty, patriotism, spiritual need, or anger.

In D&C 9:8, what burns in your bosom is a feeling that something is right. In other words, you will feel deep inside that it is right. This deep feeling could also be described as faith or a peaceful assurance. The lack of this feeling is doubt, uneasiness, or confusion, which D&C 9:9 describes as a “stupor of thought” that will cause you to turn away from, or “forget,” your wrong decision or desire.

Elder Dallin H. Oaks’ interpretation of “burning in the bosom” is similar:

What does a “burning in the bosom” mean? Does it need to be a feeling of caloric heat, like the burning produced by combustion? If that is the meaning, I have never had a burning in the bosom. Surely, the word “burning” in this scripture signifies a feeling of comfort and serenity. That is the witness many receive. That is the way revelation works. (“Teaching and Learning by the Spirit.” Ensign. March 1997)

For a discussion of D&C 9 relative to the translation of the Book of Mormon, see Did He Study it Out in His Mind?