Which sefarim are traditionally learned in Elul*, because they have a special significance to the mood and message of the month?
*Or even without a tradition, which sefarim are good to learn during this time?
I forget who told me this, but I heard that Tomer Devorah is traditionally studied in the month of Elul.
I've been told that the Bobover Rebbe told his chassidim to learn Tomer Devorah this year, and that Rav Moshe Wolfson told his Kehilla the same a few years back.
This practice makes sense, because the first chapter of Tomer Devorah is about Hashem's 13 Middos, and how they should be taken into practice, which is a focus of the month of Elul and Yamim Noraim.
(Hebrewbooks, print Hebrew, print English)
These are just examples, but you can find more of all the above types at Hebrewbooks and other seforim websites; I linked to them for illustrative purposes only, not necessarily because they are the best of each type.
Rabbi Yosef Ber Soleveitchik would teach Likkutei Torah from the Ba'al HaTanya during Elul in preparation for Rosh Hashana.
The relevant Ma'amarim start in Parshas Reih and continue from there.
I haven't been able to find someone who reports specifically what from there he would learn with his students.
I know many Yeshivas where the Rosh Yeshiva interrupts his regularly scheduled programming of a vaad or whatever public mussar seder he has to go through Shaarei Teshuva of Rabbeinu Yona during Elul. I know the Rosh Yeshiva of Passaic (R' Meir Stern) used to do this for his machshava chabura, but I don't know if he still does.
In Ner Israel, the Rosh Yeshiva (during my time there) would do this for 5 minutes after Maariv for Elul.
Shulchan Aruch HaRav writes (O.C. 429):
חכמים הראשונים תקנו בזמן שבית המקדש היה קיים שיתחילו הדרשנים לדרוש ברבים הלכות הרגל שלשים יום לפני הרגל דהיינו ... ומי"ד באלול ואילך ידרשו הלכות החג ... ותקנה זו לא נתבטלה מישראל אף לאחר שחרב בית המקדש ... והעיקר לדרוש ולהורות להם דרכי ה' וללמד להם המעשה אשר יעשון ולא כמו שנוהגין עכשיו ובדורות הללו שאין החכם שונה לתלמידיו הלכות (לפי שהכל כתוב בספר) מצוה על כל אחד ואחד שילמוד הלכות הרגל קודם הרגל עד שיהיה בקי בהם וידע המעשה אשר יעשה
Summary: Starting from the 14th of Elul one needs to learn Hilchos Sukkos from Halacha books until one is thoroughly familiar with them enough to know the proper procedure for the holiday.
One year in high school, my rebbi taught our class Hilchos Teshuva from the Rambam during the month of Elul.
I am learning Nesiv HaTeshuva, by the Maharal. Nesivos Olam is the Maharal's "mussar sefer" according to his introduction, and Nesiv HaTeshuva is the section about teshuva. It talks about the nature of Teshuva, and is therefore helpful in appreciating the significance of Teshuva.
Chovos HaLevovos – Rabbeinu Bachya ibn Paquda (Early 11th Century). Arguably the most important mussar work of all time, the Chovos HaLevovos is divided into ten Shearim (“Gates”).
For Ellul perhapes
Shaar HaTeshuva (“Gate of Repentance”) – on repentance.
Shaar Cheshbon HaNefesh (“Gate of Self-Accounting”) – on the importance of introspection and self-examination and the proper ways to do this.
Would be the most relevent.
Reb Meir Stern gives a weekly Rambam shiur. During Elul Zman he learns the Rambam on Teshuvah.