Selichot is a Hebrew word from the root s-l-ch, meaning forgiveness. It is used to refer to a number of prayers in which Jews ask for forgiveness from G-D.
There are three key times when Selichot are recited.
The Selichot on the minor fast days are quite short and are said as part of the regular morning service.
The Selichot said in the week before Rosh Hashanah are the most well known. They are either said before the morning service, or during the night before (the Jewish day runs from nightfall to nightfall, and Selichot can be said from the start of the "day" - ie nightfall the previous day, until the next morning).
In the Ashkenazi tradition, Selichot are first recited from the last Saturday night before Rosh Hashanah, although if the first day of Rosh Hashanah is a Monday or Tuesday, they are started the saturday night before. In the Sephardi tradition, they are recited for the whole month before. They are then recited until 2 days before Yom Kippur - on the day before Yom Kippur, no Selichot are recited. Many synagogues have a Choral Selichot service on the first Saturday night.
There are then Selichot on Yom Kippur, although many congretions (including Orthodox ones) don't recite them except at the first and last services of the day (Kol Nidrei and Ne'ilah). This is in line with the Artscroll prayer books.
In all cases, the Selichot are prayers asking G-D for forgiveness. There are a number of key prayers that appear.
The 13 Attributes of G-D
Merciful G-D, merciful G-D, powerful G-D, compassionate and gracious, slow to anger, and abundant in kindness and truth. Preserver of kindness for thousands of generations, forgiver of iniquity, willful sin and error, and Who cleanses. (Exodus 34:6-7)
This was the prayer G-D taught to Moses after the Children of Israel committed the sin of the Golden Calf. Therefore it is an appropriate prayer to say on the Day of Atonement, and hence it features throughout the whole day's prayers.
This is a prayer which is hard to translate directly. It consists of 24 words, each in the plural, along the lines of "We have sinned, we have slandered" etc. The fact it is in the plural is significant - every Jew is responsible for every other Jew. On Yom Kippur it is recited out loud by everybody together. It is traditional to beat the heart lightly on each word - the heart is the seat of emotion and the source of sin.
This is a powerful prayer, recited out loud in a traditional tune by the Chazzan, with each verse repeated by the congregation. The first verse means "Hear our Voice, oh Lord our G-D, have pity and compassion on us, and accept with compassion and favour our prayer". The rest of the prayer continues in the same way. It is a simple, powerful, impassioned plea for G-D to accept our prayers.