Strong cryptography or cryptographically strong are general terms applied to cryptographic systems or components that are considered highly resistant to cryptanalysis.
Demonstrating the resistance of any cryptographic scheme to attack is a complex matter, requiring extensive testing and reviews, preferably in a public forum. Good algorithms and protocols are required, and good system design and implementation is needed as well. For instance, the operating system on which the crypto software runs should be as carefully secured as possible. Users may handle passwords insecurely, or trust 'service' personnel overly much, or simply misuse the software. (See social engineering.) "Strong' thus is an imprecise term and may not apply in particular situations.
The use of computers changed the process of cryptanalysis, famously with Bletchley Park's Colossus. But just as the development of digital computers and electronics helped in cryptanalysis, it also made possible much more complex ciphers. It is typically the case that use of a quality cipher is very efficient, while breaking it requires an effort many orders of magnitude larger - making cryptanalysis so inefficient and impractical as to be effectively impossible.
Since the publication of Data Encryption Standard, the Diffie-Hellman and RSA algorithm in the 1970s, cryptography has had deep connections with abstract mathematics and become a widely used tool in communications, computer networks, and computer security generally.
This term "cryptographically strong" is often used to describe an encryption algorithm, and implies, in comparison to some other algorithm (which is thus cryptographically weak), greater resistance to attack. But it can also be used to describe hashing and unique identifier and filename creation algorithms. See for example the description of the Microsoft .NET runtime library function Path.GetRandomFileName. In this usage, the term means "difficult to guess".
An encryption algorithm is intended to be unbreakable (in which case it is as strong as it can ever be), but might be breakable (in which case it is as weak as it can ever be) so there is not, in principle, a continuum of strength as the idiom would seem to imply: Algorithm A is stronger than Algorithm B which is stronger than Algorithm C, and so on. The situation is made more complex, and less subsumable into a single strength metric, by the fact that there are many types of cryptanalytic attack and that any given algorithm is likely to force the attacker to do more work to break it when using one attack than another.