Twister is a program for building triangulations of three-manifolds, from the viewpoint of the mapping class group. The program is now maintained by Mark Bell. The easiest way to run Twister is as a module inside of SnapPy. Please see CompuTop for material on SnapPy (maintained by Nathan Dunfield and Marc Culler) and other topological software.
For those interested, the source code for Twister is available from Bitbucket. Also available there are Python scripts, using Twister and SnapPy, for generating censuses of hyperbolic manifolds.
The material below is out of date -- in particular, you can find much improved versions of the genus two census and the genus one (with two punctures) census at Bitbucket. Other census data is also available there.