PKCS#1 and PKCS#8 (Public-Key Cryptography Standard) are standards that govern the use of particular cryptographic primitives, padding, etc. Both define file formats that are used to store keys, certificates, and other relevant information.
PEM and DER are a little bit more interesting. DER is the ASN.1 encoding for keys and certificates etc., which you'll be able to Google plenty about. Private keys and certificates are encoded using DER and can be saved directly like this. However, these files are binary and can't be copied and pasted easily, so many (if not most?) implementations accept PEM encoded files also. PEM is basically base64 encoded DER: we add a header, optional meta-data, and the base64 encoded DER data and we have a PEM file.