Crack has several meanings:

1. A narrow break or opening in something that doesn't split completely apart. When a teacup gets a crack, you can see a thin line running through it but the cup stays in one piece. A crack in a sidewalk is where concrete has split slightly, creating a gap. Doors left ajar are sometimes said to be open a crack.
2. To break something so it splits or fractures without completely falling apart. You crack an egg by tapping its shell to break it open. You might crack your knuckles. When ice starts to crack underfoot, it's time to get off the frozen pond quickly. A bat can crack when it hits a baseball too hard.
3. A sudden sharp sound, like a whip cracking through the air or thunder cracking overhead during a storm. The crack of a starting pistol signals the beginning of a race.
4. To solve something difficult, like cracking a code or cracking a challenging mystery. When detectives finally figure out who committed a crime, they've cracked the case. You might also say you cracked a really hard math problem.
5. To make a joke, as in cracking a joke. Someone with a good sense of humor might crack jokes that make everyone laugh.

The phrase crack of dawn means the very earliest part of morning, when sunlight first breaks through the darkness.