Single deadeyes (or bull's eyes) are used to guide and control a line and, particularly in older vessels, to change its direction. More modern systems would use a block for this purpose but in traditional rigs with many lines to deal with, designed when blocks were relatively expensive to make, a deadeye provided an acceptable compromise.
Triple deadeyes are used in pairs; a line called a lanyard is run back and forth between them, through the holes, so that they function again much as a block and tackle would. This provides a mechanical advantage, pulling harder on whatever the deadeyes are attached to. Pairs of deadeyes are placed in the shrouds (the lines that hold up the mast), where they are used to create greater tension in the shrouds.
In recent decades, as steel wire became the prevalent material for sailboat rigging, deadeyes and laynards gave way to metal turnbuckles for tensioning the wires. More recently, however, with the advent of high-strength and low-stretch synthetic fibres, some sailboats are using synthetic rope for standing rigging, and deadeyes and lanyards are coming back into use as tensioning devices.