The Confession, or Viduy, is a key part of the Yom Kippur (Day of Atonement) services in the Jewish tradition.
In fact, Jews start specifically asking for forgiveness about one week before Rosh Hashanah (according to the eastern European Ashkenazi tradition) or about one month before (according to the Spanish & Portuguese Sephardi tradition). The prayers are all asking for forgiveness for sins you have committed against G-D - the view is that G-D can't forgive you for a sin you've committed against a fellow man (such as stealing something). For those sins, it's up to you to ask for forgiveness directly from the person concerned, and many people will try to speak to their friends in the time around Rosh Hashanah and ask them for forgiveness for anything they have done to offend them over the previous year.
The confession prayers are recited in all five services of Yom Kippur - one in the evening (Jewish days run from nightfall to nightfall) and four during the day. They are said quietly as part of the Silent Amidah service by everybody, and then again out loud and comunally by the entire congregation when the Amidah prayer is repeated.
All the confession prayers are in the plural. This is to show that sinning reflects badly on the Jewish people as a whole, and that there is collective responsibility to ask for forgiveness for the sins. Although there is a standard wording for the confession prayers, many say that if you have specific sins, you can add them in during the silent Amidah as well.
During the days of the Temple in Jerusalem, the central service on Yom Kippur was conducted by the High Priest and also involved a core confession for the entire Jewish people. Nowdays in synagogue, although we don't do the sacrafices that the High Priest's confession involved, we describe the entire service in great detail. It is also during this part of the service that religious Jews kneel in Synagogue - this is rare as we only kneel 6 times in the year (once on each day of Rosh Hashanah and twice on Yom Kippur).
In Judaism, Rabbis can't absolve a person of a sin against G-D. Neither can the person leading the prayers. Neither can the Cohen (Priest). Each person must "Make his peace" with G-D directly through prayer. We fast on Yom Kippur as part of repentance. And hope that our prayers are accepted.