Informally, "cryptographic security" describes the type of resistance a given algorithm has against attempts to decode messages encrypted by it
provable security: a proof must exist for the algorithm which shows that the underlying difficult problem has no "back doors"; in other words, an attacker has no choice but to solve the problem directly in order to break the security
provable security doesn't rely on simple obfuscation tactics that reduce the chances that a cipher will be cracked
information-theoretic security: security that doesn't rely on the limitations of current computing technology; heavily related to "information theory"
security of information-theoretic secure algorithms will not break down with the advent of new technologies, such as quantum computing