It is brought in Sefer Matei Ephraim by R' Ephraim Zalman Margolies (considered to be the authoritative handbook for the laws and customs of the months of Elul and Tishrei), that there is a custom to say an extra measure of Psalms during Elul.
R' Elimelech of Lizhensk, the Noam Elimelech writes, that sometimes, unfortunately, Tefillah (prayer) doesn't work. At that point we need something called Tehillah (praise, adoration).
This world of Tehilla is totally Light, there is no darkness, no din (judgment), only total Chesed (compassion). For this reason King David called his book Tehillim, a book of tehillos. With it one can accomplish anything, for it is pure light, a straight connection to the Holy One Blessed be He.
The sefer Elef L'Magen, a commentary on the Matei Ephraim, says one who says Tehillim on a regular basis can avoid all kinds of spiritual encumbrances, from upon himself, his wife and children, and from upon his whole community. He brings upon himself all manner of divine influence, good blessings, success and heavenly compassion.
The most common custom (minhag) is to say 10 chapters per day during the month of Elul. In this way, one completes the Book of Tehillim (150 chapters) twice which adds up to 300 total chapters recited in Elul. Three hundred is the gematria (numerical equivalent) of the word Kofer, (Chaf/20+Fay/80+Resh/200=300), which means atonement. That is one of the goals of Chodesh Elul, to secure atonement.
Many then complete the entire Book of Tehillim again during the 10 days of repentance from Rosh HaShanoh to Yom Kippur.
Some people say on day that special liturgical poem 13 middos (attributes) of mercy are recited in Selichos. There is also a prevalent custom to say the Book of Tehillim on Yom Kippur night after the evening prayers. Some even say it twice, once sitting and once standing.
The Zohar in Parshas Mishpatim (153b) says that every word in Tehillim is like a divine name. One must be therefore be extremely careful about each word's pronunciation, not to add extra sounds or to leave out syllables.
As it says in the Yehi Ratzon (may it be His will) said at the conclusion of reciting a section of Tehillim: ". . . in the merit of the Holy and pure names that I uttered, they should atone for me all of my sins and procure for me fogiveness on account of all my transgressions. . . and help me to come to complete repentance before You, and lead me to Your service, and open our hearts to learn Your Torah. . .."
May we all be able to come before Hashem this year during the Days of Awe in complete Teshuva!!