The PDB file format is explained here. My question is what "chain identifier" is exactly? What is the difference between two residues which have the same sequence number but different chain identifiers?
The "chain identifier" does exactly what it says: it identifies the polypeptide or other molecular chain. For some structures, there is only one, so you'll only see an identifier
A. However, many structures show two or more proteins bound together, or an enzyme complexed to a substrate, or a small molecule inhibitor in a binding pocket - there are many different possibilities. Each individual chain of atoms has its own identifier.
For example, 1VKX shows my favorite transcription factor NF-κB bound to the κB DNA enhancer sequence. There are four chains in the structure: two polypeptide chains, the p65 subunit
A and the p50 subunit
B, and two DNA chains
D, one for each strand (forward and reverse).