Welding involves the use of high temperatures or pressures to cause metals of two distinct parts to coalesce at the joint. A well-executed weld is at least as strong as the surrounding base metal. However, weld processes that are improperly carried out can negatively affect the base metal near the weld site and produce weaker welds. In fusion welding, some of the base metal is melted, often with a filler metal deposited to the pool of molten metal during the process. Fusion welding processes include torch welding, arc welding (comprising several variations, such as SMAW, GMAW, FCAW, and SAW), laser beam welding, and electron beam welding. Fusion welding processes require that the parts be of similar composition. For instance, you can't join copper or aluminum to steel using fusion welding. In solid-state welding, the base metal is not melted and no filler metal is added. Solid-state welding processes include the original welding process, forge welding, as well as several modern techniques, such as magnetic pulse welding, explosion welding, and friction-stir welding. Solid-state welding is much more suitable for joining dissimilar metals than fusion welding.
Simple explanation of the explosion welding process.
Brazing bonds two pieces of metal together with a braze alloy that serves as a filler metal in the joint. The braze alloy is melted during the process and bonds the parts together when it cools. Unlike welding, the base metal of the two parts is not melted or otherwise made to coalesce. Thus, braze alloys must have a lower melting temperature than the parts being joined. Brazing can be used to join different metals together, like aluminum, copper, gold, and nickel. Properly brazed joints can be very strong, though generally not as strong as welded joints.
Video showing a copper pipe being joined to a stainless steel pipe by brazing.
Soldering is similar to brazing, but is performed at lower temperatures. The filler metal used in soldering is known as a solder. The American Welding Society has defined 450 °C (840 °F) as the line between soldering and brazing (below 450 °C is soldering, above 450 °C is brazing). Solders in the past often contained lead, but these have since been mostly replaced with lead-free alternatives due to environmental and health concerns. Soldered joints are not as strong as brazed or welded joints.
Video on how to solder copper plumbing.
In summary, welded joints are strongest and typically require the most heat (except for fancy welding techniques that rely on high pressure). Metal from both parts coalesce at the welded joint. Brazing requires less heat than welding and brazed joints are not as strong as welded joints. Parts are joined together with a filler metal that melts at a lower temperature than the base metal. Soldering is essentially the same as brazing, except soldering is performed using filler metals with melting points below 450 °C and soldered joints aren't as strong.