Cryptography relies on puzzles. A puzzle that can not be solved without more information than the cryptanalyst has or can feasibly acquire is an unsolvable puzzle for the attacker. If the puzzle can be understood in a way that circumvents the secret information the cryptanalyst doesn't have then the puzzle is breakable. Obviously cryptography relies on an implicit form of security through obscurity where there currently exists no likely ways to understand the puzzle that will break it. The increasing complexity and subtlety of the mathematical puzzles used in cryptography creates a situation where neither cryptographers or cryptanalysts can be sure of all facets of the puzzle and security.
Like any puzzle, cryptography algorithms are based on assumptions - if these assumptions are flawed then the underlying puzzle may be flawed.
Secret knowledge assumption - Certain secret knowledge is not available to unauthorised people. Attacks such as packet sniffing, keylogging and meet in the middle attacks try to breach this assumption.
Secret knowledge masks plaintext - The secret knowledge is applied to the plaintext so that the nature of the message is no longer obvious. In general the secret knowledge hides the message in way so that the secret knowledge is required in order rediscover the message. Attacks such as chosen plaintext, brute force and frequency analysis try to breach this assumption