In LDS doctrine, what is the difference between the Holy Ghost and the Light of Christ? The Holy Ghost is usually described as the member of the Godhead that helps us know what is true, prompts us to do good, and generally helps us feel God's influence. I've often heard it described as something similar to a "conscience" that helps us know what is right and what is wrong. I've heard very similar descriptions for the Light of Christ. Both are available in some degree to everyone, and both are experienced less when we go against God's will. So what's the difference between the two, or are they the same thing?
Short Answer
The Holy Ghost and the Light of Christ are closely related and are often used interchangeably, even in scripture. But there are a few important differences.
The Holy Ghost, in contrast with the Light of Christ:

Is a personage.
Directs and prompts us in specific ways according to God's will.
Can work through the Light of Christ.
Is available to everyone, but is not always present to everyone. His influence is only felt when certain conditions are met.

The Light of Christ, on the other hand:

Is an influence, an essence, or a substance that infuses the universe and emanates from the goodness of Christ.
Gives us a sense of what is right and what is wrong, and influences us to do right and avoid wrong.
Can be ignited in us by the Holy Ghost.
Is always present in all of us, but becomes more difficult to perceive as we act against its influence.

Long Answer
President Boyd K Packer of the Quorum of the Twelve Apostles in the LDS Church gave a discourse that explores the light of Christ and contrasts it with the Holy Ghost. He gives a better and more authoritative explanation than I can. Here are some especially relevant excerpts:

Most members of the Church have a basic understanding of the Holy Ghost. Most have experienced its promptings and understand why the Holy Ghost is called the Comforter.
They know “the Holy Ghost … is a personage of Spirit” (D&amp;C 130:22) and a member of the Godhead (see A of F 1:1).
But many do not know that there is another Spirit—“the light of Christ” (D&amp;C 88:7)—another source of inspiration, which each of us possesses in common with all other members of the human family....
The Holy Ghost and the Light of Christ are different from each other. While they are sometimes described in the scriptures with the same words, they are two different and distinct entities. It is important for you to know about both of them....
The Light of Christ is defined in the scriptures as “the Spirit [which] giveth light to every man that cometh into the world” (D&amp;C 84:46); “the light which is in all things, which giveth life to all things, which is the law by which all things are governed” (D&amp;C 88:13; see also John 1:4–9; D&amp;C 84:45–47; D&amp;C 88:6; D&amp;C 93:9).
And the Light of Christ is also described in the scriptures as “the Spirit of Jesus Christ” (D&amp;C 84:45), “the Spirit of the Lord” (2 Cor. 3:18; see also Mosiah 25:24), “the Spirit of truth” (D&amp;C 93:26), “the light of truth” (D&amp;C 88:6), “the Spirit of God” (D&amp;C 46:17), and “the Holy Spirit” (D&amp;C 45:57). Some of these terms are also used to refer to the Holy Ghost....
Regardless of whether this inner light, this knowledge of right and wrong, is called the Light of Christ, moral sense, or conscience, it can direct us to moderate our actions—unless, that is, we subdue it or silence it....
The Light of Christ existed in you before you were born (see D&amp;C 93:23, 29–30), and it will be with you every moment that you live and will not perish when the mortal part of you has turned to dust. It is ever there....
It is important...to know that the Holy Ghost can work through the Light of Christ. A teacher of gospel truths is not planting something foreign or even new into [the learner]. Rather, the missionary or teacher is making contact with the Spirit of Christ already there. The gospel will have a familiar “ring” to them. Then the teaching will come “to the convincing of [those who will listen] that Jesus is the Christ, the Eternal God, manifesting himself unto all nations” (Book of Mormon title page)....
President Joseph Fielding Smith spoke of the teachings of the Holy Ghost and of the Spirit of Christ: “Every man can receive a manifestation of the Holy Ghost, even when he is out of the Church, if he is earnestly seeking for the light and for the truth. The Holy Ghost will come and give the man the testimony he is seeking, and then withdraw; and the man does not have a claim upon another visit or constant visits and manifestations from him. He may have the constant guidance of that other Spirit, the Spirit of Christ.”
The Spirit of Christ is always there. It never leaves. It cannot leave.
Everyone everywhere already has the Spirit of Christ, and while the Spirit of the Holy Ghost can visit anyone, the gift of the Holy Ghost is obtained “by obedience to the laws and ordinances of the Gospel” (A of F 1:3), by submitting to “baptism by immersion for the remission of sins; [and the] laying on of hands for the gift of the Holy Ghost” (A of F 1:4). It is not automatically present like the Spirit of Christ is present. This gift must be conferred by one holding authority (see A of F 1:5)....
Once a person has received that gift of the Holy Ghost and can cultivate it together with the Light of Christ, which they already have, then the fulness of the gospel is open to their understanding. The Holy Ghost can even work through the Light of Christ....
[W]hy should we marvel at the promise that the Light of Christ is in all of us and that the Holy Ghost can visit any of us?
It should not be difficult, therefore, to understand how revelation from God to His children on earth can come to all mankind through both the Spirit of Christ and the Holy Ghost....
[T]he Light of Christ can be ignited by the Spirit of the Holy Ghost, the Comforter. We are told that then “the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you” (John 14:26).