The meaning of the Mitzvah:
The purpose of blowing the Shofar is in order to arouse the person hearing the sound to do Teshuvah and change his ways. This is similar to the blow of trumpets during battle which arouse fear and trepidation in the enemy and encourages the warriors in battle. [This Teshuvah is what affects the coronation of Hashem as explained next.]
Affecting the coronation of Hashem’s Kingship:
The Gemara states “Why do we blow Shofar? In order to make Hashem rule over us.” On Rosh Hashanah Hashem’s level of Kingship over the Jewish people is held in judgment. This includes the judgment of how much revelation of His majestic nobility will be revealed below to the creations. How much love and fear will the Jew have for the coming year. The finalization of this judgment occurs with the blowing of the Shofar. The blowing of the Shofar draws down Hashem’s Kingship for the coming year and the future decrees for each and every creation. The sound of the Shofar arouses the desire and pleasure Hashem has in reigning over us. [The question however is asked as how the simple blowing of the Shofar has so much power to affect our entire future year? This is explained on a deeper level through the next teaching brought from the Baal Shem Tov.]
The inner call of the soul:
The Shofar represents the inner cry of the G-dly soul of a Jew, yearning for its attachment and return to G-d. It is similar to a child that has sinned against his father and has become distanced from him. The child was then given a reunion opportunity with his father. The child was so overcome by feeling of remorse and longing that he could not express any words in his mouth and rather a simple cry burst out from his soul. The cry of the soul touched the heart of his father and he decided to forgive and reaccept his son. The child is the Jewish soul who has become distanced from G-d within its travels to this world. It has forgotten its form of communication with G-d and hence utters a simple cry from the depths of the soul. The cry represents regret for the past and determination for the future. This cry elicits G-d’s mercy and He chooses to forgive him for the past and reunite with him. [This sound of the Shofar however is not met without challenge, and the Satan tries to muffle the sounds so Hashem does not hear the cry of His child, as will be explained next.]
The Satan prevents the Tekios from coming out properly:
It once occurred in a community that the Baal Tokeia was unable to blow properly and the Rav of the town, Rav Mechlin, told him to turn the Shofar around and follow a certain Segula of reciting the verse of Vayehi Noam into the Shofar. After doing so the Shofar began to blow properly. The Rav responded that the Satan was standing by the entrance of the Shofar, hence preventing the sounds from being heard [and the Segula removed the Satan from the area]. See Shofar Chapter 3 Halacha 3 for further details on this subject!
The low sounds of the Rebbe’s Tekios:
A famed Chassid was once asked why the Rebbe’s Tekios on Rosh Hashanah were always so low and came out with difficulty while he heard that in Elul when the Rebbe would blow Shofar for the Rebbetzin the sounds were strong and came out with ease. The Chassid, who was a heavy set fellow, answered “If you had someone like me sitting in the hole of your Shofar wouldn’t you have difficulty blowing it?” Meaning to say that when the Shofar is being blown for the Mitzvah there are evil forces acting as an impediment to the sounds.
The Shofar blows protect the Jewish people throughout the year:
Rebbe Yitzchak stated: Every year that we do not blow the Shofar on R”H we end up receiving suffering in the end of the year. The word Shofar stands for the words “Ein Satan Viein Pega Ra” which means that when there is a Shofar blown then there isn’t any damage.
The meaning of the horn
Why do we blow with the horn of a ram?
Rebbe Avahu stated: Why do we blow with the horn of a ram? Hashem said “Blow before me the horn of a ram in order so I remember the Akeida of Yitzchak the son of Avraham and consider it as if you also sacrificed yourselves by the Akeida.”
The elevation of the animal kingdom:
The use of the Shofar for the Mitzvah affects a spiritual elevation [Beirurim] throughout the entire animal kingdom.
 Emor 23:23
 Pinchas 29:1
 Rambam Teshuvah 3/4: “Even though the blowing of Shofar on Rosh Hashanah is a Gezeiras Hakasuv, nevertheless it contains the following hint “Awaken form your slumber and search your ways. Repent and return to your creator. …”; Chinuch Mitzvah 405; Tur 581; See Likkutei Sichos 4/1144
 R”H 16a
 Tanya p. 121 [Epistle 14]; Likkutei Torah Netzavim p. 106; Sefer Hamamarim 1924 p. 12; Likkutei Sichos 9/220
 See Likkutei Torah p. 32 Mamar of Ani Ledodi”
 Tanya Igros Kodesh 14 “Through the Shofar and Tefilos a new light is drawn down” “The Or is Nistalek from the night of R”H until after Tekios”.
 Admur 582/17 “From after midday of the second day of Rosh Hashanah and onwards, we no longer bless each other with the word “Techasev” being the writing of the decree has already been finalized.” The Rebbe ibid brings a letter from his father [Likkutei Levi Yitzchak Igros Kodesh p. 210] which writes that the general writing and sealing of the decree takes place on the night of Rosh Hashanah even before Tekias Shofar, although it is completion in a way of kindness and mercy is accomplished through Tekias Shofar which marks the conclusion of the writing. The Rebbe derives from this, and based on Admur 582/17, that perhaps one can wish his friend a good writing and sealing also on the second night of Rosh Hashanah, as the completion of the writing and sealing is only after Tekias Shofar of the second day.
 Siddur Im Dach “Shaar Hatekios” p. 246
 This parable is based on a teaching of the Baal Shem Tov brought in Hashlama to Keser Shem Tov sec. 108; See also Siddur Im Dach “Shaar Hatekios”; Likkutei Sichos 2/405; Hemshech Vekacha 5637 ch.70 for a Mashal of the Berditchiver; The kabalistic meaning behind the sounds of the Shofar is brought in Kaf Hachaim 585/11 and 28; Shaar Hakavanos 99; Peri Eitz Chaim 26/3; Siddur Rashash
 Siddur Im Dach “Shaar Hatekios” p. 232
 M”A 585/11; Mateh Moshe 808; Mahril; Kaf Hachaim 585/48
 This story is brought in Mateh Moshe ibid [student of Marshal] in name of one of his teachers. This story occurred at prior to the 1600’s during the lifespan of the Mateh Moshe.
 Tur 585; Kaf Hachaim 585/51
 R”H 16b
 R”H 16a
 Likkutei Sichos 14/474