Uniform circular motion involves two types of circular paths: horizontal and vertical circles. A horizontal circle has its plane horizontal with gravity acting parallel to the axis of rotation, while a vertical circle has gravity acting perpendicular to the motion. Tension in the string varies in a vertical circle, being maximum at the bottom (mv^2/r + mg) and minimum at the top (mv^2/r - mg). The discussion also highlights the challenge of maintaining a perfect horizontal circle, as the string cannot remain horizontal while the object is in motion. Understanding these dynamics is crucial for solving related physics problems effectively.