Question:

What does it mean the every Jew has a spark of the soul of Moshiach?

Answer: 

When a Jew reveals the essence of his soul and overcomes all spiritual obstacles, this is thanks to the spark of Moshiach within his soul. 

Regarding the Jewish soul, the Midrash writes:  “The soul is called by five names:  Nefesh, Ruach, Neshama, Chaya, and Yechida.”  Chassidic teachings explain that these five names are not synonyms for one another; rather, each one refers to a distinct level and characteristic of the soul.

The first four, Nefesh, Ruach, Neshama and Chaya, refer to the four soul faculties of action, emotion, intellect and will.  These four faculties are not absolute, but can be channeled in any direction, positive or negative.  One can choose to do good acts or bad, think positive thoughts or negative ones, and have desires for good or the reverse. 

Yechida, on the other hand, is absolute and unchanging.  It is a Meta soul-power that controls and directs the expression of the other four.  Yechida is the part of the soul that is always connected to the One Above.

During the course of ordinary life, the Yechida rarely makes an appearance.  However, in times of severe stress, when a Jew’s connection to G-d is about to be severed completely, G-d forbid, the Yechida suddenly expresses itself.  In the words of the Alter Rebbe, author of Tanya, a Jew neither wishes to, nor can he, be separated from the Divine.

The teachings of Kabbalah explain that all Jewish souls are sparks of one unified Collective Soul.  Every Jew’s Yechida is a ray of the unified Collective Soul.  The soul of Moshiach is this general, collective soul.

When a Jew reveals the essence of his soul and overcomes all spiritual obstacles, this is thanks to the spark of Moshiach within his soul.  This is how all of us have the power to hasten the coming of Moshiach. 

Sources:  Isaiah 52:13.  Bereishit Rabba 14:9.  Devarim Rabba 2:37.  Perush Ramaz L’Zohar, 2:43 and 3:260.  Meor Einayim, Parshat Pinchas.  Kuntres Inyana Shel Toras Hachassidus.