Chai Elul (the 18th of Elul) is the birthday of the Baal Shem Tov and the Alter Rebbe. It is well known that the Alter Rebbe would call the Baal Shem Tov zeide (grandfather), for he was a student of the Baal Shem Tov's student, the Maggid of Mezritch. As our Sages say, "whoever teaches Torah to the son of another person is considered as if he fathered him."
Yet the Alter Rebbe called the Baal Shem Tov his grandfather; he did not say "he is as if he is my grandfather." Moreover, he told R. Baruch of Mezibush, who was the Baal Shem Tov's actual grandson, that R. Baruch may have been a grandson in the material sense, but that he -- the Alter Rebbe -- was the Baal Shem Tov's grandson in a spiritual sense. He thus implied that he had received an inner dimension of the Baal Shem Tov, a link deeper than that conveyed through birth.
To explain: The word Anochi is an acronym for the Aramaic phrase meaning: "I wrote down and gave over My soul," for by giving the Torah, G-d communicated His very essence. "The righteous resemble their Creator," and communicate their essence in their teachings.
Thus in the Chassidus which he taught, the Baal Shem Tov gave over his essence, investing more of himself than he did in fathering children. The Alter Rebbe absorbed the inner dimensions of the Baal Shem Tov's teachings, and in so doing absorbed the inner dimension of the Baal Shem Tov himself. He was thus his true grandson. The essential connection between the Alter Rebbe and the Baal Shem Tov is expressed in the fact that their birthday is the same, Chai Elul.
The intent of the Baal Shem Tov's teachings was brought to consummate perfection by the teachings of Chabad. The intellectual teachings of Chabad elevate the emotional qualities. Thus Chabad Chassidus propels even simple people, unable to appreciate its intellectual sophistication, to great spiritual levels.
The Baal Shem Tov imparted the vitality of emunah (faith) to every dimension of our lives. The unique quality of faith is that it permits a connection with G-d that transcends intellect. This advantage is, however, potentially problematic, because the spiritual plane on which a person operates through faith is far higher than his level of personal awareness.
The teachings of the Alter Rebbe enable us to bridge this gap, because the Alter Rebbe explained how to bring our spiritual potentials that transcend intellect into the realm of understanding. Through the teachings of Chabad, the functioning of our minds is shaped by the G-dly power of our soul.
Through Chassidus Chabad, we can perceive the deepest spiritual lights. This will lead to the time when "the glory of G-d will be revealed, and all flesh will see..." with the coming of Moshiach; may this be in the immediate future.