Simply put, revelation is when God speaks to man. Ancient scripture includes the stories of Moses and Abraham, great Old Testament prophets who received the word of the Lord. Other prophets fill the Old Testament and the Book of Mormon, with doctrine, instruction, and examples of the results of following (or not) the revelations of God to His prophets.
Mormons believe that God has not stopped revealing His word to His children, but that He speaks to prophets today just as He did anciently. We sustain the president of our church (Thomas S. Monson today) as a prophet. Colloquially we refer to him as "the prophet". Other men, ordained as apostles in the church, are also recognized as prophets, though the president presides.
Revelation will come to the general leadership of the church for the church, but it may come to others for their own sphere of influence. A parent may receive revelation (or some might say, inspiration) for his or her family. A church leader or teacher may receive revelation specific to his or her calling.
An individual can – and is encouraged to – seek personal revelation, particularly as it relates to matters of testimony and decision making. Moroni, a prophet at the end of the Book of Mormon, makes this promise to those who read that record:
"And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost. And by the power of the Holy Ghost ye may know the truth of all things" (Moroni 10:4).
Personally I've had that very experience: upon reading the Book of Mormon (and other scripture), I've sought the confirming witness of the Holy Ghost that it's true. That witness has come for me as a feeling in my heart and peace in my mind. Sometimes it has distilled on me as I have prayed. Other times, it has come separate from the prayer, as I listen to or read something else and I realize that my prayer has also been answered.
It is possible for us to receive divine guidance in our lives – either through prophets or through ourselves.