The time between Rosh HaShanah and Yom Kippur is known as the Ten Days of Repentance. T’shuvah (repentance) enables us to become close to Hashem again. The Rambam (Hilchos T’shuvah 7:6-7) expresses that the main point of t’shuvah is rebuilding our relationship with Hashem: “T’shuvah is great because it brings a man closer to the Sh’chinah… Yesterday he was loathsome, distanced from Hashem and an abomination, but today he is beloved, a delight, close and a friend… Before, when he davened, Hashem did not listen; and when he did mitzvos, they were torn up before Him… Today, he clings to the Sh’chinah… he calls out to Hashem and is answered… and he does mitzvos that are received with joy.”
Many have the custom of taking on extra stringencies on themselves in order to develop a pattern of more meticulous religious observance. It is also customary to listen to shiurim, learn musar, and to give tz’dakah during this time period.
The Rambam in Hilchos T’shuvah (The Laws of Repentance) (2:6), says “Despite the fact that t’shuvah and crying out to Hashem are always timely, during the Ten Days between Rosh HaShanah and Yom HaKippurim it is exceedingly appropriate, and is accepted immediately, as it says, ‘Seek Hashem when He is to be found’ (Yeshayahu 55:6).” In Maseches Rosh HaShanah (18a) it explains that, “Seek Hashem when He is to be found – are the days between Rosh HaShanah and Yom HaKippurim.”
The Rambam continues in Hilchos T’shuvah (3:4) “…Every person should view himself all year as if he were half innocent and half guilty. And that is the way he should look at the world as well, as if it were half innocent and half guilty. If he would do just one sin, he would thereby tip both himself and the entire world towards the “guilty” side, and cause great destruction. And If he would do just one mitzvah, he would tip both himself and the entire world towards the “innocent” side and cause for himself and for them salvation, as it says “The righteous person is the foundation of the world” – because his being righteous tipped the world for good, and saved it.”