Brazing is more like soldering than welding; in both soldering and brazing, a lower melting point filler metal is melted to wet the surfaces of the joint and provide the joining power. Brazing rod melts at much higher temperatures than lead-tin solder and somewhat higher temps than silver solders.
Welding, on the other hand, is a joining process that melts the base metals as well as a filler metal. The filler metal is usually almost the same composition as the base metal (with differences associated with the metal loss of the welding process) and the joint is a mixture of the base metals and the filler metal. Joints can be made using some processes that do not use an additional filler metal.
Electric welding was invented in the last decade of the 19th century but didn't really become a widely used process until maybe the '20's because the joints were frequently weak and brittle. It took a while to discover all of the details to make the process really effective. Later inventions made possible welding of active metals like aluminum and magnesium.