From the second day of the month of Elul until Yom Kippur the Sepharadic custom is to recite the Selichot prayer.
Selichot is a special Tefila which inspires us to introspect, reflect on our actions and ask forgiveness from the Almighty for our mistakes and wrongdoings. Selichot is traditionally said before the morning prayer (Shacharit) although, technically, Selichot could be said also during nighttime or even during the day. Following an ancient custom in our community we don't say Selichot in between sunset and midnight. That is why those Minyanim in which Selichot is said at night will always start after (Halakhic) midnight.
The Ashkenazi Minhag is to start Selichot services the last Sunday before Rosh haShana. However, when Rosh haShana falls on a Monday or Tuesday (it can never fall on a Sunday) Selichot will begin two Sundays before Rosh haShana.
During the entire month of Elul the Ashkenazim and most Sepharadim have the custom to blow the Shofar. The Ashkenazim blow the Shofar at the end of Shacharit. Our custom is to blow the Shofar at the end of Selichot prayers, right before Shacharit . Many Sephardim (Moroccans, and North african communities) blow the Shofar while reciting the 13 attributes of mercy (vaya'abor... ).
The goal of Selichot is to prepare ourselves spiritually for the process of Teshuba (introspection, repentance) which will culminate in Yom Kippur. Imagine Yom Kippur as a spiritual marathon: a whole day consecrated exclusively to Teshuba. No one will run the marathon without a previous training. You have to prepare months in advance to be in a good shape and endure the tremendous physical effort required in the 26.2 miles NYC Marathon. Similarly, if we want to be able to focus in the sophisticated mental and spiritual process of Teshuba for a whole day we cannot come unprepared. We need an intense training, for which the Selichot is the essential part.
READ ABC's of Elul The last month of the Jewish calendar is actually the most important - serving as preparation for the High Holidays. by Rabbi Shraga Simmons, from Aish.