One of the basic concepts in understanding what is expected of us on Rosh Hashanah is to understand the significance of the name of the holiday. "Rosh Hashanah" means 'Head of the Year'. The Rabbis could have called these days, 'Beginning of the Year', or 'Start of the Year', or even 'Happy New Year.' We call this holiday 'Head of the Year' since these two days stand in the same relationship to the rest of year as the head does to the body. Just like the head is the source of the life force of the person, and afterwards this life force is distributed to each individual organ of the body, so also Rosh Hashanah has hidden within it all of the life force of the year, and from the holiday it is apportioned to each and every individual day.

...Rosh Hashanah has hidden within it all of the life force of the year...

This is the reason that we are so strongly encouraged to use each and every moment of the two days of Rosh Hashanah in a positive way, even taking the trouble to plan how we will use our time in advance, because hidden in every moment is the potential and spiritual force for each of the days of the coming year. We see this in the actual physical organs themselves; the brain in the head is much more sensitive and delicate than the other limbs and organs of the body. Similarly a person must treat Rosh Hashanah differently than he would treat the rest of the year. Only when a person's head is healthy and complete can his body be always healthy too. This is the service of Rosh Hashanah.

What are we supposed to do special when the first day of Rosh Hashanah falls on Shabbat? The Rebbe Rashab once discussed this with his son, the future Rebbe Rayatz: Preparing for Shabbat requires a happy heart. Preparing for Rosh Hashanah requires a broken heart. What does one do when Rosh Hashanah falls on Shabbat? The Rebbe answered with the Talmudic dictum, 'what is frequent takes precedence over what is infrequent'. (This is the halachic reason we put on our talit before tefillin).

Therefore, when Rosh Hashanah falls on Shabbat, Shabbat takes precedence, and we prepare with a happy heart. 'But', his son asked, 'a happy heart is incongruous with preparing for Rosh Hashanah!' The Rebbe Rashab answered that a person begins with a happy heart. A truly happy Jewish heart means that a person is happy because he is conscious of all the blessings G‑d gives him. When a person thinks deeply about what G‑d gave him, he realizes that he has yet to fulfill all his obligations to G‑d. This is the broken heart of Rosh Hashanah that comes with the happy heart of Shabbat that prepares us for this special day.

The third Lubavitcher Rebbe, called the 'Tzemach Tzedek' met with a group of 'Cantonists' — Jewish Russian soldiers who were kidnapped as young children to serve in the army, thus deprived of a Jewish education, and usually forcefully baptized. They told the Rebbe how during their army training they would do whatever they could to hold on to their Judaism. Some would try to keep kosher, or pray a bit here and there.

...the sand was like the words of Psalms, and the water was their tears that would wash their souls clean.

Many would say Psalms they remembered from their childhood, often crying, especially when doing the grueling task of polishing the buttons of their uniforms. How would they polish them? With sand and water. The Rebbe told them that the sand was like the words of Psalms, and the water was their tears that would wash their souls clean. The soldiers answered the Rebbe that a war is not won with tears, but with a marching song! This made the Rebbe very happy!

As for the second day of Rosh Hashanah, when the shofar will be blown: The Rebbe Maharash once told the Chassidim on the eve of Rosh Hashanah before the evening prayers that the blowing of the shofar is like a person shouting 'Father! Father!' The main thing is not the words 'Father, Father', rather the main thing is the shouting. On Rosh Hashanah of that year the sound of people crying could be heard everywhere.