Recursive and recursively enumerable language definition for a layman

I've come across many definitions of recursive and recursively enumerable languages. But I couldn't quite understand what they are .

Can some one please tell me what they are in simple words?

A problem is recursive or decidable if a machine can compute the answer.

A problem is recursively enumerable or semidecidable if a machine can be convinced that the answer is positive.

StackExchange Computer Science Q#7585, answer score: 20