Electrical Soldering is the basis of all modern electronics. Although integrated circuits can largely be manufactured without it, they could not be connected to anything else without soldering.
In short, solder is an allow of tin and lead, with a fairly low melting point. It bonds to most metals when melted, making it useful for permanently connecting two pieces of metal.
You will need:
The printed circuit board (PCB) that you wish to attach an electrical component to and the electrical component.
Solder. I used 63/37 rosin core solder. The tube pictured cost about $1 and has 8 feet of solder, enough for almost any project. *WARNING* Acid core solder can damage electrical component.
A soldering iron. These come in many shapes and sizes, but are ultimately, nothing more than a bit of metal, a power source and a heating element.
A wet sponge. Keep some water nearby so you can keep it wet.
Step 1: Place the Component
Printed Circuit Boards have two sides. Typically, only one side has metal printed on it.
Place the component with its leads through the appropriate holes (deciding which ones are appropriate is beyond the scope of this instruction), with the body of the component on the side without metal printing, and the long ends of the leads on the same side as the metal printing. If the component does not stay on its own, bend the leads slightly against the sides of the holes they are in.