What happens if I exceed the load my PSU is designed to deliver?
If it's a simple linear power supply it will probably either blow a fuse or overheat, break and possibly burn down your house, since most don't have active overtemperature or short circuit protection, altrough some do. This type can be recognized by the heavy transformer it inevitably contains, whose purpose is to step down the mains voltage to about 12V.

The tranformer inside a 12 V 10 A linear power supply probably looks similar to this:
 
Switching mode power supplies have become ubiquitous in recent years, since they are lighter, more compact and more energy efficient. A switching mode power supply usually has a somewhat "intelligent" control circuit that has features like short circuit protection and over temperature protection. Such a power supply will probably shut down to protect itself if you try to draw too much current. Keep in mind that the absolute cheapest power supplies from e.g. eBay are always built down to a price and in some cases lack any sort of protection.

Switching power supplies replace the chunky tranformer with a smaller, high frequency tranformer driven by transistors:
  

Summary

It depends completely on which type of power supply you have, and you should check if it has overtemperature and short circuit protection.