Now that we know what an Omer is, what does it have to do with counting?
In order to answer this question, let us look at different aspects of this counting, piece by piece:
(The Sefer HaChinuch is the source for all of the answers below.)
1) Why do we count the days and weeks from the second day of Pesach to Shavu’os?
The Sefer HaChinuch writes that the foundation of the people of Israel is Torah. Not only is the Torah the foundation of the Jewish people, but it is the foundation of the world as well, as the world was created because of the Torah. The reason we were taken out of Egypt was so that we could receive the Torah at Sinai, and fulfill its dictates. Therefore, as the sole reason for our departure from Egypt was so that we could get the Torah, our sustaining force, we were commanded to count from the second day of Pesach until the day that the Torah was given. By doing so, we can demonstrate to ourselves how great our desire is and how we long for the day that commemorates the most special occasion in our lives. This counting is similar to that of a slave or prisoner, who counts the days to his freedom with great fervor. When a person counts to a certain event or time, it demonstrates how greatly he wants to reach that point.
2) Why do we count the days that have past, as opposed to counting the days that remain between us and our goal?
We count the days that have past rather than the days that still remain, because of the emotions that a count may elicit. We want to minimize the pain that comes with the realization that there is still time that separates us from the moment we are waiting for so anxiously. Therefore, we count the days that have passed, as the knowledge that there is time behind us will bring us joy, for we are getting closer to the moment we have been waiting for.
3)Why do we begin our count from the second day of Pesach, and not the first?
The first day of Pesach is dedicated to remembering the huge miracle that occurred, that being our redemption from Egypt. Our departure was a miracle which in essence was of proportions never seen before and a clear demonstration of Hashem’s control of our destiny. We should not cause anything to interfere with the happiness this commemoration brings. Therefore, our countdown to receiving the Torah, which brings happiness as well, begins on the next day.
4)If we count from the second day of Pesach, why then do we say “X days from the _Omer_;” Why don’t we say “X days from the _second day of Pesach_?”
It is not fitting that we enumerate the days of our count by referring to the first day as “the second day of Pesach.” We instead refer to what we do on that second day, which is what makes that second day special. On the second day, the Korban Omer was brought. The bringing of the Omer provides us with a remembrance that we believe in Hashem. We acknowledge that Hashem is the one who wants us to live, and therefore provides us with sustenance in every year, so that we can serve Him and keep the Torah. The occasion of bringing the Omer (which carried this message with it) is what makes the second day special, and therefore we use the day which we bring the Omer as the reference point for our counting.