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X.509 protocols are part of the ISO Authentication framework which recommends using public key cryptography - RSA is recommended. X.509 certs contain: version, serial#, algorithm, ident, params, issuer, not-before/not-after dates, subject (e.g. user), subject’s public key (algorithm, params, key) and sig.
Alice gets Bob’s cert from a CA - she has to trust the CA to believe it’s Bob’s cert. One has to be able to following the “chain of trust” of the CA’s who have all been signed by trusted CA’s.