I'm learning RAID 5. I know it provides redundacy and improve the read/write performance by data striping and parity. The data striping means spreading data (say, a file) on several disks.

But how does the parity work exactly? mirroring? So with a RAID system, the storage capacity is one half of the real hard disk size?

Not really. Actually if you have N disks and the smaller one has size S, the total size of the array will be
Raid 5 distributes the parity information through disks. Parity information uses an exclusive OR mechanism: it stores one extra bit to reconstruct data if one or more bits are lost. You can find a good and simple explanation here. Due to this setup if a disk is lost you can recover all data, if two disk fail simultaneously all data are lost: it is a compromise between the advantages of striping and a certain degree of redundancy.