How to know if certificate is self-signed

Self-signed certificate will have identical subject and issuer fields, but a) this is not guaranteed, and b) the inverse is not true. Just because the fields have the same value that does not mean the certificate is self-signed.

Here are the outputs from one of our internal root (root-ca.crt) and intermediate certificates (ca.crt):

$ openssl x509 -subject -issuer -noout -in root-ca.crt 
subject= /C=DE/ST=Berlin/L=Berlin/O=classmarkets GmbH/CN=classmarkets CA
issuer= /C=DE/ST=Berlin/L=Berlin/O=classmarkets GmbH/CN=classmarkets CA

$ openssl x509 -subject -issuer -noout -in ca.crt 
subject= /CN=classmarkets CA/C=DE/L=Berlin/O=classmarkets GmbH/ST=Berlin
issuer= /C=DE/ST=Berlin/L=Berlin/O=classmarkets GmbH/CN=classmarkets CA

You can see that the fields are the same for both certificates, even though ca.crt has been signed by root-ca.crt:

$ openssl x509 -noout -text -in ca.crt | grep -A1 'Key Identifier'
        X509v3 Authority Key Identifier: 
            keyid:A2:2D:AF:A0:D2:64:DF:30:F1:72:39:AC:21:AF:45:D6:D4:12:19:94
--
        X509v3 Subject Key Identifier: 
            30:B0:6B:B5:56:9A:95:7C:31:4B:B2:65:95:0D:F9:EE:E8:3D:3A:C9

$ openssl x509 -noout -text -in root-ca.crt | grep -A1 'Key Identifier'
        X509v3 Subject Key Identifier: 
            A2:2D:AF:A0:D2:64:DF:30:F1:72:39:AC:21:AF:45:D6:D4:12:19:94

Note the absence of the Authority Key Identifier in root-ca.crt.

RFC3280 states in section 4.2.1.1 (emphasis mine):

The keyIdentifier field of the authorityKeyIdentifier extension MUST be included in all certificates generated by conforming CAs to facilitate certification path construction. There is one exception; where a CA distributes its public key in the form of a "self-signed" certificate, the authority key identifier MAY be omitted. The signature on a self-signed certificate is generated with the private key associated with the certificate's subject public key. (This proves that the issuer possesses both the public and private keys.) In this case, the subject and authority key identifiers would be identical, but only the subject key identifier is needed for certification path building.

So it seems to me that the Authority and Subject Key Identifiers are a much better indicator for self-signed certs than the Issuer and Subject fields. For a self-signed certificate the Authority Key Identifier will either be absent or have the same value as the Subject Key Identifier.

Yes it is true. When certificate is self-signed, then issuer and subject field contains the same value. Also, there will be only this one certificate in the certificate path.

@Vilican answer is technically correct and should do the job most of the time. But I wanted to find out if a certificate I was examining (not some particular web site) was used as a self-signed certificate or was a CA cert.

What I found out is that valid Root CA certificates have same issuer and subject. Also Extensions -> Certificate Basic Constraints indicates this is a CA as well number of allowed intermediate CAs to be signed by this one.

While self-signed certs used to secure a web site usually are not marked as CAs as well have a DNS name as CN in Subject. And/or have a list of allowed DNS names/IPaddresses under Extensions -> Certificate subject alt name.