About a week ago while I was mopping the tile floors a part of my baseboard swelled up, which I thought was weird. Today I decide to cut the bubbled part out and when I did all that came out was this fuzzy material. Any ideas on what type of baseboard this is and how do I repair the it? Thanks in advance
That looks like MDF baseboard, which is very porous and will swell when it gets wet. It looks like there was a way for the water from your mopping to wick into the baseboard (likely a nick or cut in the factory finish).
To repair the paint, you'll need to sand the swollen part down flush to the surface. If you find yourself having a hard time getting it smooth or the fuzziness doesn't go away after sanding, skim the top of the damaged area with very thin layer of wood putty and feather it out, then sand smooth.
MDF likes to suck up paint too, so I'd use an oil based primer before finish coating. It's possible that the paint might raise the surface a tiny bit and make it rough after priming, so lightly sand with 200 grit sandpaper after the primer is fully cured.
Finally, make sure that you don't have to repeat this process at some point in the future. If this is in an area where there is a good chance the floor will get wet, make sure the caulking between the baseboard and tile is sound - particularly where the grout lines run underneath it.
Use something like this after you have dressed the surface to you liking; http://www.rustins.eu/Details.asp?ProductID=773 you need something like this or the paint will soak in.
The problem with all wood based products is their hygroscopic tendancy to absorb moisture, this can result in swelling and consequential problems. This can also occur when you have climate extremes in temperature and humidity i.e. very dry days (e.g in winter) to very humid days (e.g in summer). To overcome this we use a sealer. i.e.for solid wood it will be a paint sealer. for mdf, particleboard etc, there should be a suitable veneer on both face and back. This can be a wood veneer (with sealer above) or melamine or similar. Some cheap products use only a paper based cover which does not prevent he moisture being absorbed - which is most probably the case sited here.