In an Up-down (or Sash-type) Sawmill, a single straight saw blade is held inside a wood sash (or frame) that is driven up and down by the mill. This is a picture of the Ledyard sawmill from the 1930s with the wooden sash highlighted in brown. (The lower ends of the blade and sash are below the floor in this picture.)
Up-down sawmills are also called: frame sawmill; English gate sawmill, upright saw. The straight saw blades were often referred to historically as millsaws or long saws.
Not a sawmill, of course, but a Pit Saw was used by two sawyers to convert a log into boards using muscle power. This would not be the method of choice if there is a sawmill available (unless the sawmill eliminates your pitsawing job). Note the similarity between the frame-type pitsaw and a sash sawmill.
A Muley Saw is also a type of Up-Down sawmill, but the blade is not in a frame. Rather, the straight saw blade slides between wood guides at the top and bottom. Muley saw blades are made from a heavier gauge steel than sash sawmill blades. The name may have originated from the German mühl-sage (mill-saw). See more in the Glossary entry for muley saws.