“T’shuvah” (תשובה) is usually translated as “repentance,” “penitence” or “atonement.” The latter is also often presented as “at-one-ment,” meaning being “one” with G-d.
Yet, what is “repentance?” Does it consist only of saying, “I’m sorry” or “Forgive me” or even “Please don’t punish me”?
If so, what practical purpose does it serve, if any?
To understand “t’shuvah,” we must first look at the ideal relationship between a man or woman and G-d, as depicted in the “Gan Eiden/Garden of Eden” narrative of Torah. That relationship is a warm, familiar one. G-d is continually present in Adam and Eve’s experience. They walk through a world in which all of their needs are understood and met. They are guileless, unconflicted, and secure. There’s an identification with G-d that is like both that of the infant and that of the most spiritually advanced person.
Their act of mistaken disobedience upsets the equilibrium of that relationship. As a result, they feel separation, insecurity, fear and guilt. Their offspring, growing up without an automatic sense of G-d’s presence in their lives — the presence of an ultimate Giver and Authority — even feel murderous rage and other dangerously uncontrolled impulses.
However, within G-d’s design for Creation, even preceding it, is “t’shuvah” — the possibility of a “return” (the actual meaning of the Hebrew word) to the previous, ideal relationship. “[T’shuvah/Repentance was] created even before the World.” 
“T’shuvah” first requires the simple recognition that we’re not all we should be, or that we’ve done what we should not do. Even getting to that stage can be arduous.
But having reached that stage, does saying “Sorry. My bad” suffice?
No. Even the most sincere confession is only a first step; perhaps “second,” if the first is recognition itself.
Real t’shuvah requires a change. “Change” is another implication of the Hebrew word “t’shuvah,” coming as it does from the root “shuv” which means “to turn.”
We might have done something wrong only once — as, for example, when we went to a multiplex movie theater and paid to see one movie, then sneaked into another movie without paying.
More often, “t’shuvah” involves a change in our habitual behavior. What’s more, as Rabbi Yisrael Salanter (later followed by Freud) pointed out, this in itself can require that we look at our “unconscious” motives and urges, if we would do more than create only a veneer of civility.
But at some point after acknowledging our own behavior, we must plan an alternative: “What will I do next time when confronted with the same circumstances, or the same situation?” Without a plan, how can there be change? How many times have we promised ourselves — with deep sincerity — that we won’t do that (whatever “that” is) again, only to find ourselves doing that again?
Step 1 — Recognize
Step 2 — Admit (at least to yourself)
Step 3 — Plan an alternative reaction or behavior
This might seem like a purely “rational” process, but it can take place in our prayerful conversations with G-d, too. I think that many people “converse” with G-d, even during formal liturgical prayers. The recognition of our mistakes can itself be a thought that G-d inspires in us. The “admission” can be a sharing of what G-d already knows about us with what we come to know about ourselves. The “plan,” like the recognition, can be the outcome of a spiritual back-and-forth between ourselves and G-d; to an idea that we propose, we might find ourselves suddenly inspired by a better idea. The inspiration comes from the Divine Source of perfect wisdom.
As “change,” “t’shuvah” represents a dynamic aspect of Jewish life. Putting on a tallit, lighting Shabbat candles, and so on, are “static” in the sense that they don’t change in form or performance. “T’shuvah,” however, is most identified with “change.” Even in the process of changing, the level you reached last week is only preparatory to the level you reach this week. More than preparatory, it’s a necessary increment.
Are “t’shuvah” and “musar” the same thing?
I’d say that they overlap.
“T’shuvah” can mean a change in what you’re doing from what you “shouldn’t” do to what you “should” do. “Musar” might represent an improvement in what you’re doing, even if it’s already OK. For example, to go from being impatient to patient might be called “t’shuvah.” To go from being patient in a typical range of circumstances to an even wider range might be called “musar.” But these definitions are arguable.
Every step in spiritual development is also a step towards that relationship that Adam and Eve had with G-d in Gan Eiden.
That’s how life is supposed to be.
 Pesachim 54a (Talmud)