Fire crackers have been around for hundreds of years. They consist of black powder (also known as gun powder) in a tight paper tube with a fuse to light the powder. Black powder  contains charcoal, sulphur and potassium nitrate. A composition used in a fire cracker might have aluminum instead of or in addition to charcoal in order to brighten the explosion.

Sparklers are very different from firecrackers. A sparkler burns over a long period of time (up to a minute) and produces extremely bright and showery light. Sparklers are often referred to as "snowball sparklers" because of the ball of sparks that surrounds the burning portion of the sparkler. If you look at  you can see that a sparkler consists of several different compounds:

• A fuel
• An oxidizer
• Iron or steel powder
• A binder

The oxidizer is aluminum perchlorate or barium nitrate plus aluminum perchlorate . The fuel is charcoal and sulphur, as in black powder. The binder can be sugar or starch. Mixed with water these chemicals form a slurry that can be coated on a wire (by dipping) or poured into a tube. Once it dries you have a sparkler. When you light it, the sprarkler burns from one end to the other (like a cirgarette) -- the fuel and oxidizer are proportioned along with the other chemicals in a sparkler so that it burns slowly rather than exploding like a firecracker.

It is very common for fireworks to contain aluminum, iron, steel, zinc or magnesium dust in order to create bright, shimmering sparks. The metal flakes heat up until they are incandescent and shine brightly, or they actually burn at a high enough temperature. There is also a variety of chemicals that can be added to create colors.

The shell is lauched form a mortar tube. The tube might be a short steel pipe with a lifting charge of black powder that explodes in the tube to launch the shell. When the bursting charge fires to launch the shell, it lights the shell's fuse. The shell's fuse will burn while the shell rises to its correct altitude, and then the fuse ignites the contents of the shell so it explodes.

A simple shell used in an aerial fireworks display.
The blue balls are the stars, and the grey is black powder. The
powder is packed into the center tube, which is the bursting charge.
It is also sprinkled between the stars to help ignite them.

Simple shells consist of a paper tube filled with stars and black powder. Stars come in all shapes and sizes (several of the links below list dozens of recipes for different types of stars), but you can imagine a simple star as somthing like sparkler compound formed into a ball the size of a pea or a dime. The stars are poured into the tube and then surrounded by black powder. When the fuse burns into the shell it ignites the black powder, causing the shell to explode. The explosion ignites the outside of the stars which begin to burn with bright showers of sparks. Since the explosion throws the stars in all directions, you get the huge sphere of sparkling light that is familiar at all fireworks displays!

More complicated shells burst in two or three phases. They can also contain stars of different colors and compositions to create softer or brighter light, more or less sparks, etc. Some shells contain explosives designed to crackle in the sky, or whistles that explode outward with the stars.

If you would like to learn the details of constructing fireworks, the links below (along with the patent database) contain huge amounts of useful information.