Games with right-stick combat are based around just that - you jack fools primarily by swinging the right-stick around in various ways. It sounds simple and simplistic...and it usually is!!
Right-stick combat was created and popularized by Sony's Ape Escape series, which was developed to be a "killer app" for their brand new Dualshock controller. This new style of gameplay made it stand out from other platformers, since each gadget was based around using the right analog stick in a certain way. Since then, many other companies have tried mapping combat to the right-stick in an effort to make their game more accessible, visceral, or unique; it works to varying degrees of success.
Right-stick combat is related to, and sometimes goes hand-in-hand with, dual-stick shooters, and can be interpreted as an attempt to bring that style of gameplay into 3D. However, right-stick combat, more often than not, is more melee focused than its 2D brethren, such as Gun Fight, Sheriff, or Robotron.