The process of Empadronament basically involves getting your name on your local Padró Municipal d'Habitants, which is the register of residents in a town or municipal administrative area.
By law, all residents of a municipality have to register with the Padró in the municipality where they live habitually whatever their nationality, and if you have a number of different residences you should register in the place you spend most time.
So quite simply, the main reason why you should do the empadronament is because it's a legal obligation.
However, it confers a number of rights, such as :-
In my experience, all that is required is proof of identity such as a passport and a document that proves where you live, such as a rental or purchase contract.
You should take the full documentation to your local Council or Ajuntament.
If you live in Barcelona, you need to the OAC - Oficina d'Atenció al Ciutadà - of the District where you live.