What kind of computer tools topologists/geometrists use to visualize the objects they deal with?

Here is one case study:

An impressive animation of the Hopf fibration created by Niles Johnson using only open-source tools, available for all platforms: The Python-based mathematics program Sage was used for determining the fiber parametrizations and keeping track of all the animation data. Sage provides an interface to the ray tracing system Tachyon, which produced the individual frames. These were then stitched into an animation using FFmpeg.

SnapPy is a widely used program within the geometric topology community for studying hyperbolic structures on $3$-manifolds.

Here is a YouTube video of Nathan Dunfield showing off some of SnapPy's functionality (creating Dirichlet domains, showing how to visualize the cusps, etc).