I see a lot of traces on the back that have lost their soldermask and are likely corroded through. There are also a number of vias (little holes that connect a trace from one side of the board to other) that are in the damaged areas and they might not be connecting at all.

You'll want to clean off the bad soldermask and tin up all uncovered the traces. Use lot of flux or there won't be any tinning action going on.

Thank you for the tip. How do I clean off the bad soldermask? The only thing I have used to solder were some capacitators on the mb.

You physically scrape it off until shiny copper is visible. I normally use a screwdriver for localized things and very fine sandpaper for larger areas. You'll want to get rid of as much corrosion as possible from the exposed copper or it won't tin.

I fear that will be beyond my skill, I mean what is going to happen if the tin of a track will go over another one? Something bad I think :\
If it isn't a problem i'm going to write you a pm.

That's what flux is for, it will prevent bridges from happening for the most part. You will have to make sure there's none, solder wick will help also. Any broken traces will have to be fixed with fine wire (such as one from high density IDE cables). I won't mind a PM but I am not going to teach soldering etc, there are plenty of resources out there for that.

Get a multimeter and check all the capacitors, resistors, diodes, and traces.
If you need to repair traces then use fine cooper wire and solder.
If you need to reflow memory chips then use a heat gun and no-clean flux.
The bottom of the card looks bad. Traces are exposed or missing.

If it requires allot of work then try to find a repair shop who can fix it for you.
Micro-soldering is not easy. Requires precise equipment and electronic microscope.
I will only repair electronics that don’t require a microscope as I don’t have the proper equipment at the moment. There is epoxy available to cover exposed traces.

If it's interesting to you, I suggest you find a local maker space, amateur radio club, or something similar with an environment to work with and learn about electronics fundamentals and repair.

Thank you both for the precious advices.
I think I haven't the proper equipment for do such thing by my self, the tracks are so close that even the tinner solder tip I have would be too big.
I can give a try with multimeter.
Unlucky in my area I wasn't able to find anyone who do such repair, there was a shop but it closed down 1 year ago.

this is why you need alot of flux, the tip doesn't have to be tiny only hold a blob of solder which will stick to the tracks as you run it past thanx to the flux. If you have flux, solder and iron give it a shot.

Also you can't test resistors etc that are attached to a circut with a multimeter this is bad advice, each of those parts need to be removed then tested on their own so thats probably the last thing you do, visual inspection first

have a look at this video its for drag soldering pins but shows how flux works to make solder stick to pins pads or exposed tracks

https://youtu.be/TAUHV5myhx4

Unlucky I have no flux, I never bought it because while watching tutorials they always mention to be dangerous to breath and should be used in open air.
Also I'm watching the card and where the traces seems wiped off are traces from bios chip.
While the bios screen is the one which is not corrupted: https://i.imgur.com/AH11qA6.jpg
The back of the card is in bad shape (https://i.imgur.com/2hr0jwg.jpg) but other things may just being irrelevant.
Corrupted bios screen: https://i.imgur.com/E7ol6Jk.jpg