This is what I found on the web:

Fireworks are made with black powder or what is used to be called gunpowder. Gunpowder as you know explodes but only if it is packed very tightly in a small space. In a firecracker, or the shells that explode in a firework, the gunpowder is packed very tightly. The rockets that shoot the shells into the air us also gunpowder but only one end is packed tightly. This allows the burning gases that are formed when the gunpowder burns to have a place to get out. This is usually at the bottom. The force that is created (called the thrust) moves out of the bottom downward. The reactant force in the opposite direction (Issac Newtons third law of motion) causes the rocket to go upward. So when the gunpowder burns instead of exploding it can be used to propel the rocket and when it is packed tight it makes the shell explode in the sky.

Now lets put more chemistry into this answer. Black powder (gun powder) is an explosive mixture of about 75% potassium nitrate (salpeter) that supplies oxygen and 15% charcoal, and 10% sulfur which combined with oxygen produces heat and light. Modern fireworks are composed of potassium chlorate, potassium nitrate and potassium perchlorate to supply the oxygen for combustion. Salts of various metals produce the colors seen in fireworks. Magnesium and aluminum powder add extra sparkle and flash.

To make bright colors experts add sodium to create the color yellow, copper and barium to make blue and green, strontium for red and aluminum and magnesium for white. Adding charcoal to the gunpowder makes a flaming tail that sparkles as it shoots into the sky.

How is this all put together? A firecracker is composed of black powder packed into a small roll of paper with a fuse to light the powder: Finely ground black powder is then mixed with chemicals that will create color when they burn. This mixture is loosely packed into small pellets called stars. Then the stars are packed into a shell made of pasted paper and string formed into a cylinder. A large shell can hold over 100 stars. Course black powder is then packed tightly into the shell around the stars with one end of a fuse attached inside near the stars and the other end outside the shell.