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Fossil Groups Origins III. Characterization of the sample and observational properties of fossil systems

(Abridged) Fossil systems are group- or cluster-sized objects whose luminosity is dominated by a very massive central galaxy. In the current cold dark matter scenario, these objects formed hierarchically at an early epoch of the Universe and then slowly evolved until present day. That is the reason why they are called {\it fossils}. We started an extensive observational program to characterize a sample of 34 fossil group candidates spanning a broad range of physical properties. Deep $r-$band images were taken for each candidate and optical spectroscopic observations were obtained for $\sim$ 1200 galaxies. This new dataset was completed with SDSS DR7 archival data to obtain robust cluster membership and global properties of each fossil group candidate. For each system, we recomputed the magnitude gaps between the two brightest galaxies ($Δm_{12}$) and the first and fourth ranked galaxies ($Δm_{14}$) within 0.5 $R_{\rm 200}$. We consider fossil systems those with $Δm_{12} \ge 2$ mag or $Δm_{14} \ge 2.5$ mag within the errors. We find that 15 candidates turned out to be fossil systems. Their observational properties agree with those of non-fossil systems. Both follow the same correlations, but fossils are always extreme cases. In particular, they host the brightest central galaxies and the fraction of total galaxy light enclosed in the central galaxy is larger in fossil than in non-fossil systems. Finally, we confirm the existence of genuine fossil clusters. Combining our results with others in the literature, we favor the merging scenario in which fossil systems formed due to mergers of $L^\ast$ galaxies. The large magnitude gap is a consequence of the extreme merger ratio within fossil systems and therefore it is an evolutionary effect. Moreover, we suggest that at least one candidate in our sample could represent a transitional fossil stage.