The aurora are emissions of light caused by the excitation of nitrogen and oxygen mostly by energetic (i.e., ~1 keV to few 10s of keV) electrons coming from the Earth's geomagnetic tail (i.e., anti-sunward direction). There are proton-driven aurora too, but they are fainter and more rarely observed (partly because it takes much stronger geomagnetic storms to produce enough proton precipitation into the atmosphere).
Note that electrons with this range of energies always exist in the solar wind, however the number fluxes are much lower than those that drive the aurora we all know and love. The solar wind plasma tends to have number densities of ~1-10 cm-3 and bulk flow speeds are typically ~300-500 km/s. To put that in perspective, that corresponds to a dynamic pressure of ~0.07-2.0 nPa, or more than 50 trillion times less pressure than one atmosphere at STP.
Suppose the earth wasn't surrounded by a magnetic field. How would the Aurora look like?
So if the Earth were to suddenly have no magnetic field, there would be a diffuse aurora seen (only by special instruments, as I explain below) at almost all latitudes but it would be extremely dim.
Part of the reason for the brilliance of the aurora is due to a sort of magnetic focusing during the precipitation process. So imagine taking a volume of space much larger than the Earth filled with electrons, accelerating them earthward, and then focusing them into an area that's a mere fraction of Earth's total surface area. The combination of the focused area and high fluxes results in a large contrast between the night sky and the aurora (which are actually much dimmer than a full moon).
Would we see a sky full of colored star-like lights, with the color dependent on the energies of the incident particles, the kind of particles, with which particles the incident particles interact with and the altitude at which the interactions take place?
No, not really. It would probably look more like what we see during very strong geomagnetic storms, namely, dim and diffuse light over much of the sky. However, I doubt anyone would see much as the solar wind is incident on Earth from the same direction as sun light (i.e., only daytime auroras on such an Earth).
Aurora on Mars
Mars has aurora like Earth, but it does no longer has an intrinsic magnetic field. There are surface remnant fields and it is about these that we see faint aurora. Yes, Mars does have an atmosphere, but its much more tenuous than Earth's. In fact, a neutral gas surrounding a planetary body is really all that is necessary to produce emission of light from impacting charged particles, i.e., this is basically to what the aurora amounts.
We have also observed aurora on Jupiter, Saturn, Uranus, and I think Neptune. However, it is mostly in the UV light spectrum, not visible that we observe these.
I should point out that while the loss of an intrinsic magnetic field would be very problematic, as I discussed at https://physics.stackexchange.com/a/214509/59023, it would not mean there would be no magnetic field.
There would be two types of ionizing radiation: electromagnetic radiation at UV or higher energies and energetic particles. The net result would be a very active ionosphere like that of Venus. This would create an ionopause and a bow shock similar to what Earth's magnetosphere does (the bow shock, not ionopause), but the bow shock would be much closer.
Let's assume Earth lost all of its atmosphere. Without an intrinsic magnetic field or atmosphere to ionize, the particles would mostly pass by the planet on balistic trajectories producing a wake similar to what is seen in supersonic wind tunnels. However, the solar wind does carry a magnetic field, so the particle trajectories around the blunt object would not be purely hydrodynamic (i.e., the Lorentz force still matters).