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. Or if you'd prefer an amusing (yet still accurate!) Simpsons-themed discussion of the Omer along with an Omer calendar, check out The Homer Calendar.
I even had a Girl Scouts "Campfire Badge" which required me to learn all sorts of campfire constructions shapes and rules. In those days we roasted marshmallows on sticks in the fire. Kashrut wasn't part of my life, so I could eat and roast the best marshmallows.
But now I guess ever since I was in charge of a family and laundry, I just think of the smell. I trained my kids to go straight to the bathroom after a campfire, strip and shower, lots of soap and rinse well. And don't forget to shampoo.
For some crazy reason, I can never remember why there is such a custom. I have heard and read it many times, but... smoke gets in my brain....