Soldering (pronounced sod-er-ing) is a process used to bond electrical components using metal with a low melting point. Why would you want to do this? Well, soldering creates a molecular bond that serves as a highly-conductive path between components. Twisting wires together is easy to do (and is great for prototyping), but it also makes a mediocre circuit that can be disturbed when the wires become loose.
The most commonly used solder is Rosin-core, which is 60% lead and 40% tin. It is typically formed into long strands that are coiled around a spool. The metal is very flexible and has a low tensile strength, meaning that it can be torn apart without much effort.
To heat up the solder, a soldering iron is used. Essentially, a soldering iron is a hollow metal tube with a solid pointed tip on the end. Inside of the soldering iron, one could find coils of low-resistance metal that heat up quickly when electricity is applied. When a soldering iron is plugged into a wall outlet, current flows through these coils, generating massive amounts of heat that are transferred to the tip.
Soldering irons come in a variety of sizes and power ratings. It is best to get at least a 25 watt iron for most applications and you may want a more powerful iron for soldering large gauge wires. Soldering guns are also available, which are powered with a trigger.
To solder a component, the first step is to get the component's lead in a position where it is right next to the lead you want to attach it to. This can be accomplished by twisting the wires together or, in the case of circuit board mounting, slipping the component into place through the appropriate holes on the board. Once it is in place and stable, take your soldering iron (which should already be heated) and melt a small amount of solder onto the tip. Now apply the tip of the iron to the component's lead and keep it there until the solder transfers onto the lead. If the solder does not transfer, you might want to consider melting more solder onto the tip and trying again.
Once the liquid solder is on the lead, remove the iron and allow the solder to solidify. It is important to avoid moving the lead while the solder is drying, since it may introduce fault lines in the solder and make a weak joint. After allowing the solder to dry completely, make sure that the components are connected via the solder and that they will not easily pull apart. If any extra leads are sticking out of the solder, they can be clipped off using a wire cutter.
If you are soldering high voltage components, you should make sure that the solder is a smooth ball without any sharp points. If there are sharp points (commonly known as icicles), it is best to clean the soldering iron tip and re-solder the joint. The reason for this is that sharp points in a high voltage circuit will allow coronas to form, which decrease the efficiency of the circuit.
If you are soldering components that could be damaged by the heat, you can purchase heat sinks that clip onto the lead and dissipate most of the heat before it reaches the actual component.
Other useful accessories for soldering irons include heat resistant sponges (for cleaning the tip), clamps to hold leads together, and flux.
There are some genuine safety considerations that should be taken into account when soldering. First of all, you are working with very high temperatures and it is easy to get burned. For this reason, you might want to wear gloves of some sort to protect your fingers (I have burned mine many times!). The liquid solder is also a concern since it is possible for it to bounce into your eye. A friend of mine was soldering in a school class and was cleaning the iron's tip on a sponge, but the sponge was springy and flung the solder into the air, where it landed on his eyelid. These incidents are rare, but might make you want to wear safety goggles just in case. Lastly, the fumes produced by melting solder contain lead compounds and are cancerous. Luckily, only people who do soldering for a living are usually affected and, as long as you don't go on a 72-hour soldering marathon, you should be fine.