There are two main cryptographic primitives used for bitcoin. Hashing (on which mining is based) and signatures (which ensure that only the owner of a coin can send money).

Someone who breaks the hash function used by bitcoin could cause a lot of damage (probably manipulate arbitrary places in the block chain, which means he could fake arbitrary transactions, including mining transactions). This would probably be detected sooner or later, widely pulicized, and most probably, bitcoin would need to be restarted with a different hash function.

Someone who breaks the signature algorithm could steal bitcoins by making transactions on bitcoins that are not his own. Again, little could be done except restarting bitcoin with a different signature method.

Luckily, both primitives used in Bitcoin (AFAIK these are SHA256 and ECDSA) are considered secure and were reviewed by thousands of professional cryptographers. If anyone broke them in a meaningful, practical attack, it would break a lot more things than Bitcoin.

Most probably, a break will come slowly: Someone will discover some weaknesses, then someone else will discover some more, and before there is a practical attack that could actually be abused, the Bitcoin system will be migrated to another algorithm. This will pose challenges, and if this is not done in time (e.g. because the community fails to agree on how to do it or because an attack comes suddenly), Bitcoin will probably disappear as soon as the first practical attack is executed.

It usually takes many, many years (decades) before "secure" algorithms are broken.

There is also the threat from quantum computers. If someone (e.g. the NSA) manages to build one, they have broken ECDSA. Revealing (and proving) a working, sufficiently powerful quantum computer would mean ECDSA is broken and Bitcoin needs to migrate before the first attack becomes known or it will die.