Many have the custom to stay awake and study Torah the entire night of Shavuot.
Shavuot celebrates the day when God gave us the Torah on Mount Sinai. By studying all night, we show our love and enthusiasm for this precious gift.
Another explanation is that the Jews at Mount Sinai slept late on that historic Shavuot morning! God had to "wake them up" to teach them the Torah (sound familiar?). We rectify this by staying up all night, to ensure that we won't sleep late on this day.
Staying up all night is not a halacha or a Jewish law, but rather a custom for those who feel they are physically up to it. If you won't function properly the next day, you should not stay up the whole night.