Your question deserves a serious answer concerning as it does a major focus and component of our Judaism. As a matter of fact, such authorities as the truly great Rabbi Abraham Isaac Kook (1865–1935), first Ashkenazic Chief Rabbi of British Mandatory Palestine, dealt with this question of Teshuvah -- Repentance as a primary concern of Torah Judaism; along with it Kappara-- Atonement (as with Yom Kippur)from the Divine and Forgiveness --Mechilah on the human level.
Of course, the great Maimonides (Egypt, 12th century), delineates all aspects of these matters in his 14 volume Mishneh Torah compendium on Jewish Law.
We might expect to hear of such questions primarily prior to and during the Ten Days of Repentance, the days from Rosh Hashanah through Yom Kippur, however, clearly, any time is the right time for sinners to repent and ask for forgiveness, both from the Almighty and from human beings.
This entire subject of Sin, Repentance and Forgiveness is so big that it can only be touched upon in the briefest fashion in this response.
Allow me to highlight some of the rules by means of quotations from the Rambam's (Maimonides') Laws of Repentance, Chapter One, addressing primarily the sins between a person and his/her friend, rather than between a person and his/her God.
Law (Halachah) 1. " If a person has transgressed any one of the Torah precepts, affirmative or negative, willfully or unintentionally, they must confess before God, blessed be he . . . How does one confess? He says: 'O God, I have sinned, I have done evil, I have rebelled against you and have done this . . . I regret now and am ashamed of my acts; I will never do it again.' This represents the essential part of confession. The more anyone confesses the more praise he/she deserves."
Law (Halachah) 9. "Repentance and Yom Kippur effect atonement only for sins committed against God . . . sins committed against a fellow human . . . are never pardoned unless he/she compensates his/her neighbor and makes an apology. Even though he/she made the compensation, the wrongdoer must appease the injured person and ask his/her forgiveness."
From this it is uncontestable that one must make confession of one's sin, give compensation and apologize. While one may begin with the usual formula for sinners as found in the High Holy Day Mahzor [Prayer Book], yet our recitation is still not specific to the degree where the individual recites on his/her own the actual wrongdoing or transgression. Rather it seems to fall into general categories of sins.
When it comes time for the nullification of vows -- Hatarat Nedarim on the eve of Rosh Hashanah and during the Ten Days of Repentance, there too, we state that, "According to the law, the one offering apology and requesting absolution must specify the vow. However, please note, 'my Rabbis, that it is impossible for me to specify them, for they are too numerous . . . therefore may they be in your eyes as if I did specify them."
This, I would think, places the burden upon you when receiving a sincere request for forgiveness whether from a seasoned Jew or one new to Judaism; you must forgive even without the sinner having specified his or her wrongdoings against you.
Furthermore, we are all called upon in our Judaism not to be cruel, but rather accepting, thus offering our forgiveness immediately and easily. This is capacity to forgive is highlighted by Maimonides in his exposition, to be an ingrained Jewish trait.
Let us continue reading in Maimonides' Laws of Repentance, Chapter One:
Law (Halachah) 10. "One must not show himself or herself as cruel by not accepting an apology; he/she should be easily pacified, and provoked only with difficulty. When an offender asks one's forgiveness, he/she should forgive wholeheartedly and with a willing spirit. Even if he/she has caused him or her much trouble wrongfully, they must not avenge himself or herself, he/she must not bear a grudge. This is the way of the stock of Israel [the Jewish People] and their upright hearts ...."
It may be surprising, but largely this trait of the Jewish People to be forgiving has made its way into the Jewish Daily Prayer Book --The Siddur, where many as they close their eyes to sleep each night, about to recite The Shema -- "Hear, O' Israel, God is our Lord, God is One" (Deut. 6), begin by reciting the following words, which I wholeheartedly recommend that you consider. With this I will conclude:
"I hereby forgive anyone who has angered or caused me trouble or sinned against me, whether to my body or my finances or by failing to show me honor, or in any other matter relating to me, whether under duress or willingly, inadvertently or deliberately, whether verbally or through an act. Let no one be punished because of me."