SHA-256 and SHA-512 are both secure hashes from the SHA-2 family, but they differ in output length. SHA-256 produces 256 bits (64 chars) and SHA-512 produces 512 bits (128 chars). Neither has any practical attack known to date.
The key difference is the internal operation size. SHA-256 works on 32-bit words while SHA-512 works on 64-bit words. As a result, SHA-512 can actually be faster on 64-bit CPUs, while SHA-256 has the edge in 32-bit environments.
In terms of security margin, SHA-512 offers a longer hash, but for most purposes SHA-256's 256 bits is already more than enough. A longer hash increases storage and transmission cost, so unless you have a specific need, SHA-256 is the practical default.
In short, use SHA-256 in general, and SHA-512 when you need peak performance on 64-bit servers or a longer security margin. SHA-384 is a variant that truncates SHA-512 to block length-extension attacks, and can be considered alongside them for certificates and the like.