X.509 File Extensions

.PEM - the extension is used to denote that the certificate is encoded in ASCII (Base64 Notation). There are various types of X.509v3 files and they are (usually) prefixed with a “-----BEGIN CERTIFICATE-----” line and suffixed with a “-----END CERTIFICATE-----”
-----BEGIN CERTIFICATE-----
… base 64 encoding of the DER encoded certificate
with line endings and padding with equals signs …
-----END CERTIFICATE-----
.DER - the extension is used to denote that the certificate is encoded in binary. These files may also bear the CER or CRT extension. Proper English usage would be “I have a DER encoded certificate” not “I have a DER certificate”

.CRT - extension used for certificates. may be encoded in binary or ASCII (*NIX Convention)
.CER - similar to .CRT (Microsoft Convention)
.KEY - extension used both for public and provide PKCS#8 keys. may be encoded in binary or ASCII

the only time CRT and CER can be interchanged is when the encoding type can be identical (i.e. ASCII encoded CRT = ASCII encoded CER)

OpenSSL Certificate Manipulations

for other basic openssl commands

Manipulation	PEM Encoded Cert	DER Encoded Cert
VIEW	`openssl x509 -in cert.pem -text -noout` `openssl x509 -in cert.cer -text -noout` `openssl x509 -in cert.crt -text -noout`	`openssl x509 -in certificate.der -inform der -text -noout`
TRANSFORM	PEM → DER `openssl x509 -in cert.crt -outform der -out cert.der`	DER → PEM `openssl x509 -in cert.crt -inform der -outform pem -out cert.pem`
COMBINATION	In some cases, it is advantageous to combine multiple pieces of the X.509 infrastructure into a single file. One common example would be to combine both the private key and public key into the same certificate. The easiest way to combine certs keys and chains is to convert each to a PEM encoded certificate and then simply copy the contents of each file into a new file. This is suitable for combining files to use in applications like Apache.
EXTRACTION	Some certs will come in a combined form. Where one file can contain any one of: Certificate, Private Key, Public Key, Signed Certificate, Certificate Authority (CA), and/or Authority Chain