The world does not revolve around letter size paper. Well, the office world kind of does, but in some fields, bigger is better. Wide format paper is for those large jobs, both technical and fun.
Uses
By and large, wide format paper is meant to be used in professional situations where visuals need to happen and they need to happen big. Think architects and engineers, designers and drafters. And of course, it's worth noting that you need the right sized printer to utilize large format paper. Unless you have a steady hand with the paper cutter, 18" x 24" isn't going to do you much good with standard sized ink jet printers. Big paper requires big printers called plotters; thus, sometimes wide format paper is called plotter paper.
Wide format paper can also be found in children's classrooms, most obviously as part of an art class as it's perfect for murals. History, science and math classes can also benefit from using wide format paper, because it can be used to demonstrate complicated idea chains, like time-lines or formulas.
On a Roll
With all kinds of paper, quality counts. You can find many types of wide format paper that can be chosen to suit your purposes. High quality coated papers with glossy or matte finishes are perfect for image heavy printings, like banners or even large format photos. You can also find wide format paper specifically for photo printing. Some types of are best for black and white only printing, like basic inksticking wide format paper, and large format paper made for outdoor use.
With some exceptions, wide format paper can be found in bulk, in thick, heavy rolls to be cut down for later use.