We present a broadband spectropolarimetric survey of 563 discrete, mostly unresolved radio sources between 1.3 and 2.0 GHz using data taken with the Australia Telescope Compact Array. We have used rotation-measure synthesis to identify Faraday-complex polarized sources, those objects whose frequency-dependent polarization behavior indicates the presence of material possessing complicated magnetoionic structure along the line of sight (LOS). For sources classified as Faraday-complex, we have analyzed a number of their radio and multiwavelength properties to determine whether they differ from Faraday-simple polarized sources (sources for which LOS magnetoionic structures are comparatively simple) in these properties. We use this information to constrain the physical nature of the magnetoionic structures responsible for generating the observed complexity. We detect Faraday complexity in 12% of polarized sources at ~1' resolution, but we demonstrate that underlying signal-to-noise limitations mean the true percentage is likely to be significantly higher in the polarized radio source population. We find that the properties of Faraday-complex objects are diverse, but that complexity is most often associated with depolarization of extended radio sources possessing a relatively steep total intensity spectrum. We find an association between Faraday complexity and LOS structure in the Galactic interstellar medium (ISM) and claim that a significant proportion of the Faraday complexity we observe may be generated at interfaces of the ISM associated with ionization fronts near neutral hydrogen structures. Galaxy cluster environments and internally generated Faraday complexity provide possible alternative explanations in some cases.