Final answer:
A language L is decidable if a Turing machine exists that halts for inputs in L and fails to halt for others, illustrating the machine's capability to solve a computational problem.
Step-by-step explanation:
A language L is decidable if there is a Turing machine (computer program) that halts for inputs in L and fails to halt for other inputs. In computability theory, a decidable language is also referred to as a recursive language. The concept of a decidable language is critical because it helps in understanding the limitations of what can be computed or solved by a machine. For example, a program checking a string's membership in a particular language may halt and output 'yes' if the string belongs to the language or may never halt if the string is not part of the language.