We analyze the observed properties of nested and single stellar bar systems in disk galaxies. The 112 galaxies in our sample comprise the largest matched Seyfert versus non-Seyfert galaxy sample of nearby galaxies with complete near-infrared or optical imaging sensitive to length scales ranging from tens of parsecs to tens of kiloparsecs. The presence of bars is deduced by fitting ellipses to isophotes in Hubble Space Telescope (HST) H-band images up to 10" radius and in ground-based near-infrared and optical images outside the H-band images. This is a conservative approach that is likely to result in an underestimate of the true bar fraction. We find that a significant fraction of the sample galaxies, 17%+/-4%, have more than one bar, and that 28%+/-5% of barred galaxies have nested bars. The bar fractions appear to be stable according to reasonable changes in our adopted bar criteria. For the nested bars, we detect a clear division in length between the large-scale (primary) bars and small-scale (secondary) bars, in both absolute and normalized (to the size of the galaxy) length. We argue that this bimodal distribution can be understood within the framework of disk resonances, specifically the inner Lindblad resonances (ILRs), which are located where the gravitational potential of the innermost galaxy switches effectively from three-dimensional to two-dimensional. This conclusion is further strengthened by the observed distribution of the sizes of nuclear rings which are dynamically associated with the ILRs. While primary bar sizes are found to correlate with the host galaxy sizes, no such correlation is observed for the secondary bars. Moreover, we find that secondary bars differ morphologically from single bars. Our matched Seyfert and non-Seyfert samples show a statistically significant excess of bars among the Seyfert galaxies at practically all length scales. We confirm our previous results that bars are more abundant in Seyfert hosts than in non-Seyfert galaxies and that Seyfert galaxies always show a preponderance of ``thick'' bars compared to the bars in non-Seyfert galaxies. Finally, no correlation is observed between the presence of a bar and that of companion galaxies, even relatively bright ones. Overall, since star formation and dust extinction can be significant even in the H band, the stellar dynamics of the central kiloparsec cannot always be revealed reliably by the use of near-infrared surface photometry alone.