transgression TRANSGRES'SION, n. The act of passing over or beyond any law or rule of moral duty; the violation of a law or known principle of rectitude; breach of command. He mourned because of the transgression of them that had been carried away. Ezra 10. Forgive thy people all their transgressions. 1 Kings 8. 1. Fault; offense; crime.

SCRIPTURES and QUOTES:
Moses 5:

 10 And in that day Adam blessed God and was filled, and began toprophesy concerning all the families of the earth, saying: Blessed be the name of God, for because of my transgression my eyes are opened, and in this life I shall have joy, and again in the flesh I shall see God.
 11 And Eve, his wife, heard all these things and was glad, saying: Were it not for our transgression we never should have had seed, and never should have known good and evil, and the joy of our redemption, and the eternal life which God giveth unto all the obedient.

1. Partaking of the forbidden fruit was not a sin

To help explain that Adam and Eve did not sin when they partook of the forbidden fruit, read the following statement from Elder Dallin H. Oaks:

“It was Eve who first transgressed the limits of Eden in order to initiate the conditions of mortality. Her act, whatever its nature, was formally a transgression but eternally a glorious necessity to open the doorway toward eternal life. Adam showed his wisdom by doing the same. …

“… We celebrate Eve’s act and honor her wisdom and courage in the great episode called the Fall. … Elder Joseph Fielding Smith said: ‘I never speak of the part Eve took in this fall as a sin, nor do I accuse Adam of a sin. … This was a transgression of the law, but not a sin.’ …

“This suggested contrast between a sin and a transgression reminds us of the careful wording in the second article of faith: “We believe that men will be punished for their own sins, and not for Adam’s transgression” (italics added). It also echoes a familiar distinction in the law. Some acts, like murder, are crimes because they are inherently wrong. Other acts, like operating without a license, are crimes only because they are legally prohibited. Under these distinctions, the act that produced the Fall was not a sin—inherently wrong—but a transgression—wrong because it was formally prohibited. These words are not always used to denote something different, but this distinction seems meaningful in the circumstances of the Fall” (in Conference Report, Oct. 1993, 98; orEnsign, Nov. 1993, 73).

PERSONAL DEFINITION:

Transgressing is merely acting contrary to a said law. Transgressing only is related to sin when someone acts contrary to what is morally right. Adam and Eve did not act against God's will when they partook of the fruit, for it was His will that they should do so. They only acted against the said law. They did not do something wrong, because before they transgressed, there was no good or evil.
I think Transgression is also possible in ignorance, because sinning in ignorance isn't really sinning, if you have no knowledge of what is right or wrong.
Mistakes happen, and are part of our growth here on earth, and so are sins, but we are accountable for our sins.
The more truth we learn, the less mistakes we are apt to make, and the less sins we will want to make. :)