The Advanced Camera for Surveys (ACS) Fornax Cluster Survey is a Hubble Space Telescope program to image 43 early-type galaxies in the Fornax cluster, using the F475W and F850LP bandpasses of the ACS. We employ both 1D and 2D techniques to characterize the properties of the stellar nuclei in these galaxies, defined as the central luminosity excesses relative to a Sersic model fitted to the underlying host. We find 72+/-13% of our sample (31 galaxies) to be nucleated, with only three of the nuclei offset by more than 0.5 from their galaxy photocenter, and with the majority of nuclei having colors bluer than their hosts. The nuclei are observed to be larger, and brighter, than typical Fornax globular clusters, and to follow different structural scaling relations. A comparison of our results to those from the ACS Virgo Cluster Survey reveals striking similarities in the properties of the nuclei belonging to these different environments. We briefly review a variety of proposed formation models and conclude that, for the low-mass galaxies in our sample, the most important mechanism for nucleus growth is probably infall of star clusters through dynamical friction, while for higher mass galaxies, gas accretion triggered by mergers, accretions and tidal torques is likely to dominate, with the relative importance of these two processes varying smoothly as a function of galaxy mass. Some intermediate-mass galaxies in our sample show a complexity in their inner structure that may be the signature of hybrid nuclei that arose through parallel formation channels.