Rosh Hashanah Orthodox Union
Some of the meanings and ideas to focus on while listening to the blowing of the Shofar in shul are:

• Proclaiming the coronation of G-d as King of the Universe
• Awakening our minds from a state of spiritual sleepiness
• Reminding us of the shofar heard at Mount Sinai, when we accepted G-d's Torah
• Appealing to G-d with a simple, primal cry from the depth of the human soul. Indeed, as we ahve seen, the  "Teruah" sound of   the  Shofar  is based on the sound of a cry.
• Sounding the call of the Shofar that will be heard with the coming of Mashiach, who will redeem us, G-d willing, soon

Two blessings are said over the blowing of the Shofar. One blessing is over the Mitzvah  (Torah Obligation) itself, which concludes "...Who has commanded us to hear the sound of the Shofar."   From the text of the blessing we see that the Mitzvah is to hear  the shofar.  Therefore, if you see someone blowing the Shofar, or even blow it yourself, you have still not fulfilled the mitzvah if you do not hear the sound itself.  An example of this is if one blew the Shofar into a chamber such that the actual sound is drowned out by the echoes.The second is the She'Hecheyanu Blessing which is said to thank G-d for, simply, giving us life, and allowing us to reach this moment, and this new experience, of hearing the sound of the Shofar  in the New Year.
There are four basic parts to Teshuvah:

1. Leaving the Sin
2. Regret
3. Confession Before G-d
4. Acceptance for the Future
1. Leaving the Sin
Leaving the sin consists of stopping the commission of the sinful act. One cannot do Teshuvah if one continues to do the sin, even if he or she were to perform the next three steps perfectly.
2. Regret
Regret consists in sincerely regretting one's wrong action. One must be genuinely ashamed and embarrassed over one's sins.
3. Confession Before G-d
Acceptance for the future consists of resolving in one's heart never to commit the sin ever again.
4. Acceptance for the Future
Confession before G-d consists of an oral confession spoken out loud, in which one formulates in words the commitments and attitudes one has reached in his or her heart. One should say, "I have sinned, I have done such and such; I deeply regret my actions, and I declare before G-d, Who knows my innermost thoughts, that I will never do this sin again."
But...
1. The above steps only work for sins committed against G-d; for sins committed against other people, one must first ask forgiveness from that person before G-d will accept the Teshuvah.

This is the source of the practice by many Jews to contact all of their family, friends and co-workers during this period to ask for forgiveness for anything we may have done to upset them during the past year(s).

2. These four steps are of course only valid if we do Teshuva AFTER THE FACT. One cannot say in advance - "I can do this sin, then do Teshuvah and He will forgive me..." It simply doesn't work that way as it may in other belief systems.
On the other hand...
One should keep in mind that Teshuvah is an ongoing process that cannot be accomplished overnight. No matter how many times a person may stumble in the Teshuvah process, that person has to simply pick him or herself up and keep trying to stay on the right path.
What G-d is really looking for is the sincerity of the effort that a person puts into their Teshuvah!
The First Rosh HaShanah
On the 25th day of Elul the world came into being (Midrash - Jewish Lore).

Adam, the First Man, and Chava, the First Woman, were created on the last of the Six Days of Creation, the Eve of Shabbat  - This day is celebrated as Rosh HaShanah.
Thus, when we say on Rosh HaShanah "Today is the Birthday of the World," we mean that it was the Date of Creation of the first Human Beings.
The Midrash (VaYikra Rabbah 29:1) tells us that on the very first Rosh HaShanah Adam and Chavah committed the very first sin. G-d judged them and forgave them. G-d said, "Just as you were judged before me on this day and emerged forgiven, so will your children be judged this day and emerge forgiven."
For this reason, G-d sees fit to judge all the people and nations of the world every year on this day.
Conclusion & Summation
Let us consider the following selection as an appropriate summation of all we have learned about Elul and Teshuva:
"When you really get down to it, teshuvah today is a kind of death and rebirth: a demise of the past and a birth of a new life and a new creature. There is a severing with the previous "me" and the creation of a new "me" who has a new awareness, a new sensitivity, new ambitions and dreams and longings.


              
"Somehow, the connections to the new life become very powerful, and at the end of the trail, the rewards are enormous: a sense of having returned home and of being part of our majestic tradition, a sense of the grandeur and beauty and warmth of it all, the awareness of God's presence in one's daily life, the feeling of meaning and purpose that permeates one's self. 
"Life becomes coherent and whole again."