According to the Torah (Lev. 23:15), we are obligated to count the days from Passover to Shavu'ot. This period is known as the Counting of the Omer. An omer is a unit of measure. On the second day of Passover, in the days of the Temple, an omer of barley was cut down and brought to the Temple as an offering. This grain offering was referred to as the Omer.
Every night, from the second night of Passover to the night before Shavu'ot, we recite a blessing and state the count of the omer in both weeks and days. So on the 16th day, you would say "Today is sixteen days, which is two weeks and two days of the Omer." The Orthodox Union has a chart that provides the transliterated Hebrew and English text of the counting day-by-day.
The counting is intended to remind us of the link between Passover, which commemorates the Exodus, and Shavu'ot, which commemorates the giving of the Torah. It reminds us that the redemption from slavery was not complete until we received the Torah.