I know some people here hate when people use definitions in talks, etc., but in some cases I think it's very important to do so if only because then we're all using the same definition at that particular time or at least the audience knows what definition the speaker is using. From the LDS Bible Dictionary, Repentance:
The Greek word of which this is the translation denotes a change of mind, a fresh view about God, about oneself, and about the world. Since we are born into conditions of mortality, repentance comes to mean a turning of the heart and will to God, and a renunciation of sin to which we are naturally inclined.
That's probably pretty close to my own belief. Regular dictionary definitions are also close to this definition. In essence I think repentance means to be willing or desirous to change or turn toward God. Generally speaking I think anytime we encounter the word repent in scripture we can substitute the word change.
The Bible Dictionary doesn't include an entry for penance, of course, because penance isn't part of LDS theology - except the way repentance is often talked about seems more like penance to me. So for the sake of being on the same page, here's the definition of penance from dictionary.com that pretty much matches what I think penance is:
1) a punishment undergone in token of penitence for sin.
2) a penitential discipline imposed by church authority.
That said, here are some quotes I have used in talks on the subject of repentance. They center on the Greek word metaneo:
From Theodore M. Burton, Seventy, in the Ensign (there is also a slightly longer version which was a BYU devotional). Like all of these talks/articles I don't agree with every word and they all have elements of penance included. This first one is my favorite.
https://www.lds.org/ensign/1988/08/the- ... e?lang=eng
...The Old Testament was originally written in Hebrew, and the word used in it to refer to the concept of repentance is shube....
...can you think of a kind, wise, gentle, loving Father in Heaven pleading with you to shube, or turn back to him—to leave unhappiness, sorrow, regret, and despair behind and turn back to your Father’s family, where you can find happiness, joy, and acceptance among his other children?
That is the message of the Old Testament. Prophet after prophet writes of shube—that turning back to the Lord, where we can be received with joy and rejoicing. The Old Testament teaches time and again that we must turn from evil and do instead that which is noble and good. This means that we must not only change our ways, we must change our very thoughts, which control our actions.
The concept of shube is also found in the New Testament, which was written in Greek. The Greek writers used the Greek word metaneoeo to refer to repentance. Metaneoeo is a compound word. The first part, meta-, is used as a prefix in our English vocabulary. It refers to change. The second part of the word metaneoeo can be spelled various ways. The letter n, for instance, is sometimes transliterated as pn, and can mean air, the mind, thought, thinking, or spirit—depending on how it is used.
In the context in which meta- and -neoeo are used in the New Testament, the word metaneoeo means a change of mind, thought, or thinking so powerful that it changes one’s very way of life. I think the Greek word metaneoeo is an excellent synonym for the Hebrew word shube. Both words mean thoroughly changing or turning from evil to God and righteousness.
Confusion came, however, when the New Testament was translated from Greek into Latin. Here an unfortunate choice was made in translation; the Greek word metaneoeo was translated into the Latin word poenitere. The Latin root poen in that word is the same root found in our English words punish, penance, penitent, and repentance. The beautiful meaning of the Hebrew and Greek words was thus changed in Latin to a meaning that involved hurting, punishing, whipping, cutting, mutilating, disfiguring, starving, or even torturing! It is no small wonder, then, that people have come to fear and dread the word repentance, which they understand to mean repeated or unending punishment.
The meaning of repentance is not that people be punished, but rather that they change their lives so that God can help them escape eternal punishment and enter into his rest with joy and rejoicing. If we have this understanding, our anxiety and fears will be relieved. Repentance will become a welcome and treasured word in our religious vocabulary.
My second favorite is this one, also a BYU speech and also a Seventy (Weatherford T. Clayton, not to be confused with his brother L. Whitney). Besides repentance, this one has some other gems - I also like the rock thing. I especially like the last paragraph of this quote.
https://speeches.byu.edu/talks/weatherf ... -redeemer/
How do we come to Him? We begin with faith in Christ, and central to true faith is action. When we have faith in Jesus Christ, we want to act according to that faith. The Book of Mormon prophet Moroni called this having “real intent.” For example, if we have faith in Christ, we come to church each Sunday. If we have faith in the Savior, we pay our tithes and offerings. If we have true faith in our Redeemer, we do something about it, because we have real intent. And every time we act according to our faith in Him, we follow Him by hearing and “doing” His words. As we act with faith in Him, the Lord confirms our faith by blessing us with peace, answers to prayer, direction, comfort, and joy. And thus our foundation becomes stronger, wider, and deeper.
As we act on His words, we are doing something called repenting. In the New Testament, repentance comes from the word metanoeó, from the words metá and noeó, meaning “to change one’s mind or purpose.” Isn’t that interesting?
Every time we turn more to Christ, we are repenting—we are following Him. When we sincerely pray to the Father, in a very real sense we are repenting. When we read the scriptures and ponder them, we are repenting. As we make changes because of what we are learning about Christ and His gospel, we are repenting. When we do things that make us better, kinder, gentler, more sensitive, more spiritual, more virtuous, and truer, we are repenting. Whenever we choose the better path, we are repenting. Though we all repent of things that are sinful in our lives, most of our repenting comes from hearing His words and doing them—from turning to Him. This builds our foundation, and we want that foundation to be as big and as wide and as deep and as sturdy as possible.
And Elder Nelson also seems to understand the concept of metaneo in April 2007 GC (and he also has a great handle on penance):
https://www.lds.org/general-conference/ ... n?lang=eng
The doctrine of repentance is much broader than a dictionary’s definition. When Jesus said “repent,” His disciples recorded that command in the Greek language with the verb metanoeo. This powerful word has great significance. In this word, the prefix meta means “change.” The suffix relates to four important Greek terms: nous, meaning “the mind”; gnosis, meaning “knowledge”; pneuma, meaning “spirit”; and pnoe, meaning “breath.”
Thus, when Jesus said “repent,” He asked us to change—to change our mind, knowledge, and spirit—even our breath. A prophet explained that such a change in one’s breath is to breathe with grateful acknowledgment of Him who grants each breath. King Benjamin said, “If ye should serve him who has created you … and is preserving you from day to day, by lending you breath … from one moment to another—I say, if ye should serve him with all your whole souls yet ye would be unprofitable servants.”
Hope that helps.