Doctrine & Covenants 112:23-26 reads (emphasis mine):
Verily, verily, I say unto you, darkness covereth the earth, and gross darkness the minds of the people, and all flesh has become corrupt before my face. Behold, vengeance cometh speedily upon the inhabitants of the earth, a day of wrath, a day of burning, a day of desolation, of weeping, of mourning, and of lamentation; and as a whirlwind it shall come upon all the face of the earth, saith the Lord.
And upon my house shall it begin, and from my house shall it go forth, saith the Lord; first among those among you, saith the Lord, who have professed to know my name and have not known me, and have blasphemed against me in the midst of my house, saith the Lord.
If I read this prophecy correctly, everyone on earth has become corrupted (or unclean, see 1 Nephi 10:21, etc.) including members of the LDS Church, that this corruption will be the justification for vengeance, and that vengeance will begin with the Church and then proceed to the rest of the Earth.
Which, to simplify, sounds like the Lord will clean His house1 before He cleanses the world.
What do the scriptures2 and/or Church leaders suggest is the corruption the Lord is speaking about?
1 This revelation was received in 1837, after the 1836 dedication of the Kirtland temple. It could be argued that "house" refers specifically to the Kirtland temple, or temples in general. However, setting aside the fact that the LDS Church no longer owns the Kirtland temple and presupposing the Lord had more in mind that a single geographical location when discussing the need to cleanse people from sin, I'm electing to interpret "My house" as "My church." If scripture or Church leaders suggest otherwise, I'm open to correction.
2 The LDS Scriptures include the Holy Bible, the Book of Mormon, the Doctrine & Covenants, and the Pearl of Great Price.