The empadronamiento is the registration of your residency with the municipality where you live. By law, anybody living in Spain should be registered (empadronado) in the city where they reside.
Registering is an easy process. You just need to go to city hall with your passport and a document proving your residency, usually a copy of your house deed (escritura), a rental contract or some utility bills. Call your ayuntamiento to ask for the exact documents needed.
Registering is beneficial as it confirms your residency in the community and simplifies a variety of administrative procedures such as school inscription, health benefits...
The Spanish football federation also requires a certificate of empadronamiento to deliver a license (ficha) to foreign players participating in the national divisions.