What is a digital certificate, what guarantees its trustworthiness, and what's the dominant standard?

A certificate binds a public key to an identity; a digital signature (from a CA) guarantees its authenticity and integrity; the standard is X.509.

• Binds a public key ↔ an identity (e.g. "this key belongs to www.hslu.ch").
• That binding is itself digitally signed by a Certificate Authority, so its authenticity and integrity can be verified — tampering would break the signature.
• The near-universal format is X.509.

So a certificate is essentially a signed statement: "CA X attests that public key K belongs to identity I."

Tip: Every HTTPS padlock you click shows you an X.509 certificate — issuer, subject, validity dates, and the public key.