Does .pem file contain both private and public keys?

You can decode your PEM formatted x509 certificate with the following command:

openssl x509 -in cert.pem -text -noout

PEM certificate contains public key only or private key only or both.

---

For the following example:

-----BEGIN CERTIFICATE-----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-----END CERTIFICATE-----

you will get:

Certificate:
    Data:
        Version: 3 (0x2)
        Serial Number: 0 (0x0)
    Signature Algorithm: ecdsa-with-SHA256
        Issuer: C = BE, O = GnuTLS, OU = GnuTLS certificate authority, ST = Leuven, CN = GnuTLS certificate authority
        Validity
            Not Before: May 23 20:38:21 2011 GMT
            Not After : Dec 22 07:41:51 2012 GMT
        Subject: C = BE, O = GnuTLS, OU = GnuTLS certificate authority, ST = Leuven, CN = GnuTLS certificate authority
        Subject Public Key Info:
            Public Key Algorithm: id-ecPublicKey
                Public-Key: (256 bit)
                pub:
                    04:52:d8:8d:23:8a:e3:67:d7:86:36:b1:20:0b:09:
                    7d:c8:c9:ba:a2:20:95:2f:c5:4a:63:fa:83:5f:ce:
                    78:2f:8f:f3:62:ca:fd:b7:f7:80:56:9d:6e:17:b9:
                    0e:11:4c:48:b2:c0:af:3b:59:17:16:30:68:09:07:
                    99:17:fe:dd:a7
                ASN1 OID: prime256v1
                NIST CURVE: P-256
        X509v3 extensions:
            X509v3 Basic Constraints: critical
                CA:TRUE
            X509v3 Key Usage: critical
                Certificate Sign, CRL Sign
            X509v3 Subject Key Identifier: 
                F0:B4:81:FE:98:12:BF:B5:28:B9:64:40:03:CB:CC:1F:66:4E:28:03
    Signature Algorithm: ecdsa-with-SHA256
         30:45:02:20:31:ae:c0:3d:4a:3f:21:be:85:17:fc:f0:c7:b2:
         31:07:2a:38:56:43:d1:36:d5:95:e1:7e:52:c0:06:43:87:a7:
         02:21:00:97:8c:0e:b8:3c:0a:41:af:ae:a5:cf:06:7e:d5:c4:
         d8:2f:ff:e2:62:80:34:10:ba:22:dd:35:81:46:93:22:9a

To understand difference between Public Key Algorithm and Signature Algorithm sections read this (both are public).

A PEM file may contain just about anything including a public key, a private key, or both, because a PEM file is not a standard. In effect PEM just means the file contains a base64-encoded bit of data. It is called a PEM file by allusion to the old Privacy-Enhanced Mail standards which preceded S/MIME as a mail security standard. These standards specified the format of various keys and messages in a particular base64 format. See RFC 1421 for example.

Typically a PEM file contains a base64 encoded key or certificate with header and footer lines of the form -----BEGIN <whatever>----- and -----END <whatever>----. Over time there have evolved many possibilities for <whatever>, including private keys, public keys, X509 certificates, PKCS7 data, files containing multiple certificates, files containing both the private key and the X509 certificate, PKCS#10 certificate signing requests, ...

RFC 7468 has been written to document this de facto format.