The cabinet on my bathroom had some water damage on one of the sides. I sand it and also used a chisel to remove the damaged area. The material that I removed looks like 
.
This is how it looks the surface after sanding it: 
. Should I apply wood filler, and then sand it before prime/paint, or there is another way to fix this? I don't even know what type of wood this is. Could you clarify this too?
This sort of fluffy, fibrous material indicates MDF or HDF (medium-density or high-density fibreboard, e.g. hardboard) rather than particleboard/chipboard. The clue to the nature of these materials is right there in their names, with the first two made from wood fibre and the third from particles or chips.
From your second picture it looks like there's a thin layer of this fibre-based material with a wood veneer under it, which suggests MDO (Medium-Density Overlay, a type of plywood) except that MDO should be waterproof and is unlikely to be badly affected by some exposure to water — MDO is popular for exterior signage.
So there's a chance this is just a thin sheet of standard MDF or HDF glued to a substrate of plywood.
Should I apply wood filler, and then sand it before prime/paint, or there is another way to fix this?
If you're happy with the flatness then you could prime and paint now; obviously you want to use waterproof or at least water-resistant paint. Less obviously, you need to take care to seal the bottom edge, where any standing water has plenty of time to wick into the edge. If this bottom edge had been sealed well initially it's possible the damage would never have occurred.
If you need to make the surface more level you can of course fill, but two cautions:
Many wood fillers are not made for this kind of fill work and won't spread well, and additionally won't adhere strongly in large (and thin) applications like this.
It's challenging to get a large fill good and flat, especially in situ, no matter what filler material you use!
Because of 1. it seems to be very common to fill shallow defects (over a large area) using fillers not originally intended for use on wood, such as automotive fillers like the original Bondo. Note that many or most such fillers are two-part products which require blending a small amount of hardener into a larger mass of putty.
Agree completely with the answer by @graphus but to suggest an alternate approach: skin the side of the cabinet with something new and avoid all the sanding/filling/etc. The only detail you’d have to figure out is what kind of a piece of wood you’d want to put on the raw front edge. Prime paint all the hidden edges and surfaces. Apply whatever finish you want to the visible face. Caulk the joints where water might pool.
I’d still prime paint the old stuff thoroughly to help with future problems.
