Context. Virialized halos grow by the accretion of smaller ones in the cold dark matter scenario. The rate of accretion depends on the different properties of the host halo. Those halos for which this accretion rate was very fast and efficient resulted in systems dominated by a central galaxy surrounded by smaller galaxies that were at least two magnitudes fainter. These galaxy systems are called fossil systems, and they can be the fossil relics of ancient galaxy structures. Aims: We started an extensive observational program to characterize a sample of 34 fossil group candidates spanning a broad range of physical properties. Methods: Deep r-band images were obtained with the 2.5-m Isaac Newton Telescope and Nordic Optic Telescope. Optical spectroscopic observations were performed at the 3.5-m Telescopio Nazionale Galileo for ~1200 galaxies. This new dataset was completed with Sloan Digital Sky Survey Data Release 7 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 (Δm12) and the first and fourth ranked galaxies (Δm14) within 0.5 R200. We consider fossil systems to be those with Δm12 ≥ 2 mag or Δm14 ≥ 2.5 mag within the errors. Results: 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 the fossil systems are always extreme cases. In particular, they host the brightest central galaxies, and the fraction of total galaxy light enclosed in the brightest group galaxy is larger in fossil than in non-fossil systems. Finally, we confirm the existence of genuine fossil clusters. Conclusions: Combining our results with others in the literature, we favor the merging scenario in which fossil systems formed from mergers of L∗ 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 fossil group candidate in our sample could represent a transitional fossil stage. This system could have been a fossil in the past, but not now owing to the recent accretion of another group of galaxies.