Ok, gonna give my .02 cents here…
COILSPRINGS: believe it or not, their true purpose is to the absorb shock from the road and provide ride comfort. performance springs are stiffer and improve handling by reducing body roll during cornering, and by lowering the car’s center of gravity (lowering springs/coilovers). really good springs reduce body roll without sacrificing too much comfort (they do this by varying the stiffness at different parts of the spring...this is called a progressive spring rate)
SHOCK ABSORBERS/STRUTS: “shock absorber†is a misnomer (wrong name). The true purpose of the shock absorber or strut is to dampen the rebound of the springs…and therefore keep the wheels glued to the road. Performance shocks are designed to take more heat build-up than oem shocks, and are filled with gas rather than liquid. Also, performance shocks are usually stiffer to improve responsiveness, keeping the car better glued to the road…and to some extent, reduces body roll during cornering. Adjustable shocks let you pick a soft comfortable ride or a hard ride for better handling.
CONTROL ARMS/TRAILING ARMS: these babies keep the wheels and shock/strut assemblies moving in a nice straight up-and-down path. Without them, the shock assemblies would rip themselves out of their retaining bolts after a few bumps.
ANTI-SWAY BARS: these are usually called “sway bars†for short. See pissedoffsol’s description, cause I don’t know much about these.
STRUT BARS/TIE BARS: these are stiff braces that tie opposing strut assemblies together. Installing them significantly reduces the flex in your chassis during hard cornering and over bumps. This makes steering quicker and makes handling more nimble.
That about covers it I think