What are the current explanations and models for the formation of a central bar in some galaxies, and what is the difference that can produce a barred galaxy instead of a spiral one ?
It's my impression from skimming the literature just now that much of the current activity in this field is based on N-body simulations. It's hard for me as an outsider to evaluate the analytical scenarios that are being tested, but I think that the key idea is that certain Lindblad resonances which excite density waves are the main instabilities that lead to bar formation. See also section (4) of this page, which gives a broad overview of stellar dynamics in galaxies.
In short, there are several guesses out there. See this paper by Steinmetz for an overview of the understanding of galaxy shapes in general in 2002.
The key citation in the wikipedia article linked to by Cem is a 2002 paper by Bournard and Combes (free on the arxiv) which describes gravitational N-body simulations which follow processes of bar formation and re-formation. They find a strong dependence on the frequency of bar formation on the distribution of mass in the galaxy:
A galactic disk may either keep a single bar or get several successive bars. By simulating several disks, we study the dependence on physical parameters. We find that the two determining parameters are the bulge to disk and halo to disk masses ratios. Light bulge galaxies maintain their bar, while bars are destructed and regenerated in heavy bulge galaxies. The halo mass has also an effect on the disk evolution.
Another paper from 2002 by Steinmetz and Navarro does a different simulation and observes that a bar is formed in their model galaxy due to the tidal forces from a satellite galaxy. Here's a related image from their paper:
Caption: Fig. 4. The tidal triggering of bar instability by a satellite. At z=1.6 the disk develops a well defined, long-lasting bar pattern as a result of tidal forcing by the satellite shown in the z=1.2 panel. The satellite is finally disrupted at z=1.18 after 7 pericentric passages. The bar pattern, however, survives for several Gyr, as shown in the picture. Stars less than 1.5 Gyr old are shown in blue, older stars in red. Note that the bar is most prominent in the young component. Horizontal bars in each panel are 5 (physical) kpc long.
Another proposed mechanism for the formation of a bar is that the dark matter halo surrounding the galaxy is elongated rather than spherical. This is unlikely to apply to massive galaxies where the stellar disc is massive enough to locally overwhelm the gravitational potential of the halo, but in dwarf galaxies the disc is (almost) a collection of tracer particles in the halo potential. The orbits in the disc end up elongated in the direction of the longest halo axis, resulting in what is basically a bar. It's a bit different from the photogenic bars you see in pictures of giant galaxies - these are much weaker visually and are easier to pick out looking at the velocities of stars/gas in the disc. The bar also rotates much more slowly (low pattern speed); its rotation speed is locked to the rotation of the halo, which is much slower than the orbital period of the galaxy. This model is still a "prediction"; there is at best tentative evidence for the presence of weak, slow bars in dwarf galaxies, but it is not ruled out, either. The details are laid out in this paper (I am a co-author). Here's part of one of the figures, showing the elongated stellar distribution, i.e. the bar, in a simulation (left), the corresponding elongation in the dark matter halo (centre) and the elongation of the gravitational potential (right).