Stop using the laptop immediately for safety's sake. Batteries have been known to explode. Remove the battery and dispose of it safely. 2 years is a long time for batteries. Have the computer checked for damage.

We don't see any pictures in your post. Only you can see it. 

https://imgur.com/gallery/jJXHIaL

This is the image I uploaded. Also you have to be kidding me, 2 years is a long time for batteries?

I have 5 years old laptops and their battery life cycle is normal. I regularly check my alienware battery cycle and it was normal last I checked. This has to be something else. 

I already called an electrician to come and check the power on the sockets today. 

That battery appears to be bulging.  Get rid of it -- immediately.

Can I continue to use the laptop without the battery? Like just plugged in without it?

Ok I have dismounted the battery apart. It is the weirdest battery I've seen cured in halfs, but it doesn't seems to be bulging.

http://imgur.com/aMQ9ASP

The white box connection seems quiet well also, I think it is the DC IN one that is burning. Any suggestions?

And why on earth is the battery **bleep** on to the motherboard!?

The battery looks OK from the side.  That said, what's happening is that the system (and battery) are likely overheating because you're blocking the air intake by using the system in your lap.  The system should be used on a hard, flat surface to allow cooling air access to the interior.

If there's no evident damage to the system, reinstall the battery and see if using it properly (i.e., NOT in your lap) reduces the temperature.

Nice. Just one more question, is it safe to keep charging it with those cables? I mean, is that burn just a sign of overheating or is it a sign of a damaged cable that need replacing.